I'm actually surprised by how many people don't actually seem to care about their teams...
I love all my teams, though some more than others (especially one team in Emerald that I lost (I still regret) and my Diamond team.)
My Diamond team have fully integrated themselves into existence and assimilated their personalities, and I'd NEVER give them up. Ever.
Maybe it's because I've never been a competitive battler, maybe some people just don't use their imagination to make the pokemon MORE than just pixels, maybe it's because I'm just completely insane (seriously. You won't find a friend of mine who denies that), but I find my Team, and I don't let them go. Sometimes there'll be a gap in my team for a long time (even if other pokemon are caught, they're just deposited), but once their place is taken, no other will ever claim it.
Also, just recently, with my Pearl game (in which I have a party of six and a team of three*) I somehow grew vastly attatched to one of my HM-slaves (a bibrael), which is... truly shocking, as I generally think that they're really stupid looking. She's just... different to the other bibrael now (even if she looks just as stupid). It's the first time that's happened to me... mind you, it's also the first time that I've constantly carried one around with me and actually trained it, but even at the point that I started training her (when she was a bidoof. Bibrael learns more HMs) I was growing attached…
(* read: I carry six pokemon, but only three ever get used in battle- three team members, two HM-slaves, one pokemon that I'm too lazy to deposit (Besides, it's been there longer than any, save my starter)(also, I just find it amusing to walk around with an arceus in my party)