My personal example is vampires. One of my first DMs allowed a guy to chamge the LA of a vampire to 3 because the weaknesses of vampires are so prohibitively awful. Due to this early influence on me ive also revised their LA for PCs.
Any time a LA was significantly more than the starting Hit Dice, it was a bad investment regardless of anything extra the race brought with it. I've unfortunately only seen examples of not doing adjusting it, which lead to very weak characters, or stupidly strong characters that died to a single attack from a "level appropriate" enemy. LA +5 but only 2 hit dice sort of thing and doing very poorly in a 6th level encounter. Problematic.
LA was always a very ill judged and broken mechanic that invariably produced a lack of balance. Fundamentally, the math pretty much always fails to work in all cases.
The most common source of a race or template having LA is unbalanced stats. For example, if a race or template grants a bonus to ability scores like STR, DEX, or CON, then it's clearly superior to other races.
LA tries to fix this by applying a constant penalty or modifier to the character's level. But this fails in all cases.
For low level characters, the loss of one or more levels always costs more than the benefits gained by having higher attributes. As a simple example, a 2nd level character is slightly more than twice as capable than a 1st level character, but the unbalanced attributes of a +1LA character never make a 1st level character as capable as a 2nd level character. This is increasingly true of higher numbers like +3 or +4 LA.
But at higher levels, the situation reverses. Consider the simple case of an 18th level fighter. It's fairly easy to show that there exists some amount of ability gain for which a player would willingly trade one or more levels of fighter for the increased ability. For example, +2 CON grants that 18th level fighter 1 hp/level, so trading down to 17th level while trading up for +2 increased CON pretty much invariably results in more hit points rather than less. A +2 bonus to STR grants you a +1 bonus to hit and damage as well as advantages in other areas, which is in many cases better than +1 BAB. A +2 DEX bonus grants you bonuses to initiative, missile combat, dodge bonus to AC, and many skills - which is generally flat out better than a feat. Add to these sundry benefits like natural armor bonuses, and very quickly it's obvious that the high level character (with the exception of pure casters trading for a template that didn't provide spell abilities) is getting a big benefit by switching out HD for racial bonuses. The larger these bonuses, the more impactful they are and the higher level the character the less as a percentage they are giving up by giving up a few HD.
As a result what you tended to see is that characters that were forced to begin play as low or mid-level characters generally optimized by taking low +LA races or templates, while characters that could begin play at high level were generally optimized by taking high +LA races or templates.