So, I recently got a couple of my pets to level 60 (woot, woot! XD), but after that, they don't seem to go to level 61. (One of my pets now has more than the required experience, but hasn't leveled up.) Am I doing something wrong?

Nope. There's a level cap which is 60 right now until new exploration content comes out. :)

Oh. :( That's incredibly weird, but thanks.

Well, it's not that weird if you think about it. You wouldn't want to let players have their pets at levels way higher than the opponent, or there won't be much challenge in exploring and battling. And the level slowly increases for each new region or zone released.

So since the admins want to work on improving current features on the site, they don't have the time to code and release new regions just yet, so it's quite logical to put a cap on pets' levels so the current regions still pose some challenge with battling; and when the new regions do get released, it won't be an easy overkill for the players. =P

That's kinda what new pets are for - and giving higher-level pets minimal XP for defeating opponents that are weaker than them, and requiring them to gain more XP for higher levels, etc...

The admins want the user's pets to increase in level as the story progresses. You have followed the story just as far as it has gone at the moment. Once the next part comes out, the max level will in increases again :)