Sort of.
Wizards gained spell levels at levels 1, 3, 5, 7, 9, 11, 13, 15, and 17; sorcerers at levels 1, 4, 6, 8, 10, 12, 14, 16, and 18; bards at levels 1(2), 4, 7, 10, 13, and 16. The spell restrictions for a bard were cantrips at 1st level and one level behind for 1st-level spells compared, the same as sorcerers for 2nd-level spells, and back to only 1 level behind for 3rd-level spells. They were close until 4th-level spell access but pointing at that specifically ignores the other points I brought up.
All 3 examples cast spells at the caster class level.
3e did not use standard spell levels. 3e adjusted spell levels to the class for when they wanted the class to access it, so what happened was a bard would gain access to spells WotC thought were bard spells at similar levels to other full spell casting classes.
The other major point is bards had magical songs. At 1st level that included countersong, bardic inspiration, and fascinate. Spell like ability with songs replaced other slots. They later gained inspire competence, suggestion, inspire greatness, song of freedom, inspire heroics, and mass suggestion. That doesn't mean they needed a stunted spell progression. Their magic was simply split into multiple mechanics.
Here are the numbers of spells known for the relevant comparison:
Level | Wizard | Sorcerer | Bard | Paladin | Ranger |
1 | 5 | 6 | 6 | 1 | 0 |
2 | 7 | 7 | 9 | 2 | 0 |
3 | 9 | 8 | 12 | 2 | 0 |
4 | 11 | 10 | 14 | 3 | 1 |
5 | 13 | 12 | 16 | 3 | 1 |
6 | 15 | 14 | 17 | 4 | 2 |
7 | 18 | 17 | 20 | 4 | 2 |
8 | 20 | 20 | 21 | 5 | 3 |
9 | 22 | 23 | 22 | 5 | 3 |
10 | 24 | 24 | 25 | 6 | 4 |
11 | 26 | 28 | 26 | 6 | 4 |
12 | 28 | 29 | 27 | 7 | 5 |
13 | 30 | 32 | 30 | 7 | 5 |
14 | 32 | 33 | 31 | 8 | 6 |
15 | 34 | 36 | 32 | 9 | 7 |
16 | 36 | 37 | 36 | 10 | 8 |
17 | 38 | 39 | 38 | 11 | 9 |
18 | 40 | 40 | 40 | 12 | 10 |
19 | 42 | 42 | 42 | 15 | 13 |
20 | 44 | 43 | 43 | 16 | 14 |
That includes the most songs in with spells known for the bard, and the smite and lay on hands abilities for the paladin. Spell casting ability scores are assumed 19, 16, and 14 as the minimums for all spell levels in each class.
Here are the number of magic actions based on those numbers:
Level | Wizard | Sorcerer | Bard | Paladin | Ranger |
1 | 5 | 9 | 3 | 1 | 0 |
2 | 7 | 11 | 6 | 2 | 0 |
3 | 9 | 12 | 8 | 2 | 0 |
4 | 11 | 17 | 11 | 3 | 1 |
5 | 13 | 18 | 14 | 4 | 1 |
6 | 15 | 23 | 16 | 5 | 2 |
7 | 18 | 25 | 18 | 5 | 2 |
8 | 20 | 30 | 21 | 6 | 3 |
9 | 22 | 32 | 23 | 6 | 3 |
10 | 24 | 36 | 24 | 8 | 4 |
11 | 26 | 38 | 27 | 8 | 4 |
12 | 28 | 42 | 29 | 9 | 5 |
13 | 30 | 44 | 30 | 9 | 5 |
14 | 32 | 48 | 34 | 10 | 6 |
15 | 34 | 50 | 37 | 12 | 7 |
16 | 36 | 54 | 39 | 13 | 8 |
17 | 38 | 56 | 43 | 14 | 9 |
18 | 40 | 60 | 46 | 15 | 10 |
19 | 42 | 62 | 49 | 18 | 13 |
20 | 40 | 64 | 51 | 20 | 14 |
This includes songs and smites, and a pool of healing for lay on hands.
Here is the caster levels (based on class level).
Level | Wizard | Sorcerer | Bard | Paladin | Ranger |
1 | 1 | 1 | 1 | 0 | 0 |
2 | 2 | 2 | 2 | 0 | 0 |
3 | 3 | 3 | 3 | 0 | 0 |
4 | 4 | 4 | 4 | 2 | 2 |
5 | 5 | 5 | 5 | 2 | 2 |
6 | 6 | 6 | 6 | 3 | 3 |
7 | 7 | 7 | 7 | 3 | 3 |
8 | 8 | 8 | 8 | 4 | 4 |
9 | 9 | 9 | 9 | 4 | 4 |
10 | 10 | 10 | 10 | 5 | 5 |
11 | 11 | 11 | 11 | 5 | 5 |
12 | 12 | 12 | 12 | 6 | 6 |
13 | 13 | 13 | 13 | 6 | 6 |
14 | 14 | 14 | 14 | 7 | 7 |
15 | 15 | 15 | 15 | 7 | 7 |
16 | 16 | 16 | 16 | 8 | 8 |
17 | 17 | 17 | 17 | 8 | 8 |
18 | 18 | 18 | 18 | 9 | 9 |
19 | 19 | 19 | 19 | 9 | 9 |
20 | 20 | 20 | 20 | 10 | 10 |
A bard gaining an 8th-level wizard spell as a 6th-level bard spell removes the significance of the spell levels, and bard song DC's being based on either perform check or class level brings the DC's in line as well.
The only thing a person does by looking at a specific part of the class is miss the big picture on how it functions overall. 3.5 bards are full casters when we look at how spell levels works and also recognize that they achieve that because of the separate spells and songs mechanics.
It's obvious that when a system is being created looking at past editions that separates spell casting tables the way 5e did that bards are far more in line with full casters as well.
I might want to compare various classes and other aspects instead of just wizards. For example, here's the caster level comparison:
Level | Priest | Wizard | Bard | Paladin | Ranger |
1 | 1 | 1 | 1 | 0 | 0 |
2 | 2 | 2 | 2 | 0 | 0 |
3 | 3 | 3 | 3 | 0 | 0 |
4 | 4 | 4 | 4 | 0 | 0 |
5 | 5 | 5 | 5 | 0 | 0 |
6 | 6 | 6 | 6 | 0 | 0 |
7 | 7 | 7 | 7 | 0 | 0 |
8 | 8 | 8 | 8 | 0 | 1 |
9 | 9 | 9 | 9 | 1 | 2 |
10 | 10 | 10 | 10 | 2 | 3 |
11 | 11 | 11 | 11 | 3 | 4 |
12 | 12 | 12 | 12 | 4 | 5 |
13 | 13 | 13 | 13 | 5 | 6 |
14 | 14 | 14 | 14 | 6 | 7 |
15 | 15 | 15 | 15 | 7 | 8 |
16 | 16 | 16 | 16 | 8 | 9 |
17 | 17 | 17 | 17 | 9 | 9 |
18 | 18 | 18 | 18 | 9 | 9 |
19 | 19 | 19 | 19 | 9 | 9 |
20 | 20 | 20 | 20 | 9 | 9 |
Wizards got bonus XP for casting spells to overcome a problem, for crafting magic items, and for successfully researching new spells. Bards got XP based on the value of treasure, the the HD of monsters defeated, and every time they used a special ability.
Level | Cleric | Druid | Wizard | Bard | Paladin | Ranger |
1 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | 1500 | 2000 | 2500 | 1250 | 2000 | 2250 |
3 | 3000 | 4000 | 5000 | 2500 | 4000 | 4500 |
4 | 6000 | 7500 | 10000 | 5000 | 8000 | 9000 |
5 | 13000 | 12500 | 20000 | 10000 | 16000 | 18000 |
6 | 27500 | 20000 | 40000 | 20000 | 32000 | 36000 |
7 | 55000 | 35000 | 60000 | 40000 | 64000 | 75000 |
8 | 110000 | 60000 | 90000 | 70000 | 125000 | 150000 |
9 | 225000 | 90000 | 135000 | 110000 | 250000 | 300000 |
10 | 450000 | 125000 | 250000 | 160000 | 500000 | 600000 |
11 | 675000 | 200000 | 375000 | 220000 | 750000 | 900000 |
12 | 900000 | 300000 | 750000 | 440000 | 1000000 | 1200000 |
13 | 1125000 | 750000 | 1125000 | 660000 | 1250000 | 1500000 |
14 | 1350000 | 1500000 | 1500000 | 880000 | 1500000 | 1800000 |
15 | 1575000 | 3000000 | 1875000 | 1100000 | 1750000 | 2100000 |
16 | 1800000 | 3500000 | 2250000 | 1320000 | 2000000 | 2400000 |
17 | 2025000 | 500000 | 2625000 | 1540000 | 2250000 | 2700000 |
18 | 2250000 | 1000000 | 3000000 | 1760000 | 2500000 | 3000000 |
19 | 2475000 | 1500000 | 3375000 | 1980000 | 2750000 | 3300000 |
20 | 2700000 | 2000000 | 3750000 | 2200000 | 3000000 | 3600000 |
I can make it easier for you. By the time the bard reaches 20th level outside of XP bonuses the cleric is about to hit 18th level, the druid is still 14th level, the wizard is about to turn 16th level, the paladin is about to turn 17th level, and the ranger has just turned 15th level. Here are your spell slots and caster levels for each:
- cleric -- CL 18; 8/8/8/8/6/4/2 and will also never have access to 8th or 9th-level spells. Possibly only 5th-level spells.
- druid -- CL 14; 6/6/6/5/3/2/1 and will also never have access to 8th or 9th-level spells. Possibly only 5th-level spells.
- wizard -- CL 16; 5/5/5/5/5/3/2/1 and is the only class with up to 9th-level spells. Possibly only only 4th-level spells.
- paladin -- CL 9; 3/3/3/1 and will never have better than 4th-level spells.
- ranger -- CL 8; 3/3/2 and will never have better than 3rd-level spells.
- bard -- CL 20; 4/4/4/4/4/3 and will always have up to 6th-level spells.
That wizard possibly has access to more spells prepped, yes; but the bard has a much better caster level than any other class in an edition where that matters. I say possibly because 2e also used ability score requirements for spell levels and it takes a lot of luck or generous rolling methods for those other spell casters to meet those requirements.
It's facetious to point to levels for spells in 3e as an argument and ignore the fact that clerics and druids wouldn't be either applying that same standard because that's the first edition either had those spell levels.
We also cannot ignore the 2e bard's ability to force a paralyzation save at will for influencing attitudes outside of combat.
I would point out that treasure XP bonuses and special ability spam are more effective sources of XP for faster level increases than the table suggests. My bards were often 2-3 levels higher than wizards at lower levels and more after name level. 2e bards made better fireballs.
As I mentioned above, it's pretty clear that a system separating spellcasting tables into 2 charts (full or half) the bard would fall into the full caster looking at paladins and rangers for comparison.
I demonstrated that bards did not have reduced caster level progression. Caster level was equal to bard class level.
Gaining certain spells at different levels is what enabled bards to learn spells at similar levels to other full casters. That has no bearing on the wizard list just like having a bard spell list instead of using the wizard list has nothing to do with a class's status as a caster. I gave more earlier in this post as well.