Thinking of a bit more backstory for Lips (as well as Trumpet Girl)....
Lips was often shy and took trumpet lessons as a way of overcoming shyness. He met Trumpet Girl at some kind of horn instrument get-together, maybe it was trumpeteer camp, maybe it was a jazz night, maybe it was something else, but they hit it off and decided to date a little.
While Trumpet Girl wasn't exactly shy, she didn't see the others at the Muppet Theater as friends like everybody else did. She played trumpet in the orchestra for money but didn't really form real friendships or have a big desire to be a big-time musician. She actually came to the theater wanting to be a chorus girl, but figured since she could play the trumpet she might as well play at the theater, and Nigel saw enough potential in her to offer her a regular job in the orchestra, even if Kermit never thought she was good enough for solo numbers. But not too long after, Trumpet Girl decided that she didn't like the experience of being a chorus girl, and was content to stay in the orchestra. Perhaps it was the fact that her original goal wasn't to play trumpet in the orchestra combined with her bad experience in the chorus line that she lacked a desire to be a big-time musician.
But after dating, she decided to talk Nigel into hiring Lips into the band. While he impressed everyone, Nigel didn't really want two musicians in the orchestra playing the same instrument (never mind the fact that the orchestra has occasionally had two whatnots playing the trumpet together), but since she did know how to play the trombone as well while Lips couldn't play any other instruments (or didn't like other instruments as much), she agreed to start playing the trombone.
Shortly after, The Electric Mayhem became impressed enough with his skills to ask him to be part of The Electric Mayhem. Trumpet Girl was never upset that they never asked her while they asked this new guy - after all, she didn't want to be a big time musician, she wasn't a big fan of rock and roll, and the band never considered her because she seemed so square - while Lips had that great rock star look while also playing the trumpet a lot better than she did. But they did break up shortly after this. Perhaps she was a little envious (suddenly wanting what she didn't really want just because somebody close to her had it), or maybe she was worried that she'd be too in the spotlight dating a member of The Electric Mayhem. But the break-up didn't really lead to much resentment from them, they just interacted with each other less than before - about as much as anybody communicated with either. And with Trumpet Girl not really hanging out with the other Muppets outside of theater gigs (at least until 2011, though most of that was at the theater), we just saw more of Lips than of her.
But while Lips proved a great asset to the band, his shyness complicated things. He loved playing the trumpet with the band and getting his solos, but he was always afraid to do his own trumpet number, and turned down offers to appear in Gonzo's trumpet gags. Lips was often set to perform at big gigs for the band, only to have shyness attacks at the last minute and chicken out. He'd often be afraid to pose for photos or and always turned down opportunities for interviews (all three of them, actually!), and his habit of not going on at the last minute led to the band refusing to acknowledge him in interviews and promotion, not wanting to risk saying that he'd be there only to be proven wrong later.
And he'd go through periods in shyness therapy, keeping him out of the band and other Muppet projects, with his biggest therapy period being from 1992-2009. He also entered therapy halfway through production of Muppets Most Wanted, which explains why he seems to be seen quite a bit in the first hour but by the end seems to have just disappeared (aside from being in the orchestra) for the rest of the film (not even being at the wedding or in the closing number). In fact he also spent a few sudden days in therapy during production of The Great Muppet Caper, which explains why he's absent from a few group scenes that otherwise include all the primary characters (I've often noticed that when they get to the Mallory Gallery he seems to disappear and reappear from the group).
But his last shyness therapy sessions have improved greatly, now he's talking a lot more (even if it's just an hour a day) and he's gone 40 days without canceling an appearance at the last minute (an all-time high for him - and he's only had one gig during this time).