The best of the bunch...
Mouse Guard: You have to have Level in successes, and Level-1 failures, in significant uses of a skill to have it go up. There is a limit of 24 skills per character (of a list of 50+).
MegaTraveller: Unlike early editions of Classic, it has both in-play experience (use the skill enough, it goes up, but you can only gain a particular skill so fast... and it also has the Int+Edu limit, where if your total of those two stats is the limit to the number of skill levels you may possess; if you exceed that with a new level gained, you must reduce another.
Classic Traveller in later revisions of the second edition, namely The Traveller Book and Starter Traveller, and in Book 7: Merchant Prince, the Int+Edu limit is applied at the end of mustering out. It is from these sources that MegaTraveller gets that rule. The 1st Edition (1977-1980 printings) and 2nd Edition (1981 and later printings of Books 1-3) of CT do not include the Int+Edu limit.
A few that other people think do, but I disagree, and why:
Classic Traveller: There is almost no skill gain in play; it takes characters years to acquire a skill by study, and years out of play to acquire one by taking a sabbatical. There is an upper limit on skills based upon the sum of Intelligence + Education, starting with certain later supplements; it's not in the little black books versions of the core rules.
Burning Wheel: At least in Revised, the skill loss rules have been deleted... Luke's commented on so doing. The rest of it, however, isn't bad, tho' in-play skill gains don't mirror in character generation. The lifepaths are a source of both inspiration and drama, but I'd not call them realistic.
SevenSidedDie notes that the original Burning Wheel rules included skill loss rules, and that those may be ported to Revised.
GURPS: While GURPS 3E has some wonderful practice requirement rules, it has no skill acquisition process in the first place. There is the official statement that 1 point = 250 hours of practice...
Skill loss Rules? Are they important?
You'll note my reasons for BW are the lack/loss of skill loss rules. Doesn't make it any less of a great game (in fact, I'm in a skype game of BW) but it does make it less realistic. I've personally lost a number of skills from lack of use, and many characters probably should have...