I don't consider it right that archers are restricted by ammunition when melee types aren't. My philosophy is until everyone has to pay some kind of per-hit upkeep on their weapons, no one has to.
I don't know which edition you play but..
In any edition with Vancian, and to a lesser extent any edition without, nearly all classes and specialties run out of ammo or have a "per-hit upkeep" for their attacks. In 4e if you had Encounter abilities you had a "per-hit upkeep" to track. If you had dailies then you had them too.
If you were in an edition with vancian magic then you would run out of spells. Running out of spells and not running out of arrows seems like an odd contrast.
In fact the only resource you typically don't run out of pretty much ever is a nice sharp stick. It keeps stabbing. As the butcher said in Wanted, 'you never need to reload and it never runs out of bullets' (paraphrased of course). That is the BENEFIT of using blades over ranged attacks.
The benefit of ranged attacks is that they have to range attack you back, or close the distance to attack you. You can get into cover and avoid attacks much more easily.
Saying that you don't want to have to count arrows is fine.
Saying that you shouldn't have to because swords never run out of swings is silly.
As far as the true topic however, it is less "does it ever happen" and more "should it be possible" to which I certainly answered yes.
It has happened in our games that we have had to track ammo, food, water and all other kinds of resources. Magic and gold made that much easier or more redundant at higher levels (in 3.5) but I dislike that about magic and money. I would like resource management to be something considered at all levels of play. I just don't want to be bogged down by it.
There are a number of solutions to hit the middle ground in that case, some proposed here and some proposed other places. Having well stocked inventories or access to vendors are certainly solutions. Being able to make arrows (food, etc.) is a solution too. The problem arises when magic can override this need entirely or when there IS no need at all. I find it a little ridiculous to think that characters may not need to buy arrows in 5e but then buying marbles/caltrops to trap passageways.
As far as the whole "at-will" part - I don't like at-will and it is something I tried then disallowed for the playtest. I don't have a problem with a wizard using a crossbow if they feel they have nothing left to contribute. Never have. I don't see why they get unlimited access to unerring missiles when no one else does. Especially when they can cast sleep and other area effect spells. It doesn't make them useless it just means that they have to be careful and not waste their resources frivolously. I would expect the same kinds of things from non-casters. I want my martial characters to be more tactical in their thinking instead of just charging in full force as well. I encourage creative and intelligent gameplay, as well as roleplaying in my games. Not having to worry about truly minor things (TP) is fine but I don't appreciate when they are rendered completely moot (unlimited arrows).