I play at a bluegrass jam every Sunday night here in Charlotte, NC. There are two guitar players who are good, sometimes excellent, lead players, but they can't keep steady rhythm if you held a gun to their heads (and fellow CoMando Joe Cline and I have both been tempted at times). I just don't understand how this could be. But the problem points out that rhythm deserves some serious practice attention. I guess I'm talking mainly about chording rhythm, but I'm sure the need holds true for all kinds of music, including classical pieces that have a mandolin part that requires steady accompaniment.
Surprise, surprise, use a metronome! All of the advice about building up speed and steadiness on tunes applies to rhythm playing. It may seem boring to play chords along to a clicking metronome, but you should put in some time on it to bang the beat into your head and fingers. Also good to play along with records--you'll figure out real quick when you're lagging or rushing.
As you learn a new tune, spend some time learning and practicing the chords (if you play a kind of music that requires chordal accompaniment). I'm amazed at pickers who can "play" a tune but don't know the chords. I would say they're not REALLY playing the tune. If it's a song with lyrics, play and sing (with a metronome). (Even if you don't sing in public, you can sing in private, and doing both is excellent for coordination and learning.)
Learn a variety of chords rather than just a few chop shapes. We've discussed this recently on here, and others have rightly pointed to Niles Hokkanen's little chord book and Jethro's books on three-note chords. (If you can play some of Jethro's chord solos, you're really a rhythm AND a lead player. ) Chords are a fascinating part of music if you move beyond the basics (as fundamental and important as the basics are; unlike others who have made disparaging comments about the good old chop, I'd say for a bluegrass mandolinist, there's nothing more important or satisfying).
Take pride in your rhythm playing, as much as in lead playing. The mandolin is an important rhythm instrument in a number of styles of music. People say the banjo defines bluegrass, but I really think the drive of the mandolin does. Even if you can't play lead or improvise, you can be a valuable addition to any bluegrass group by playing solid, steady, driving rhythm. Listen to a great rhythm player (Mr. Monroe, Sam Bush, Tim O'Brien, to name just three) and learn what they're doing. It's clear that they've put as much thought, time, and pride into their rhythm playing as they have their leads. And I'll argue that you ought to do it first! Don't be a flashy soloist who can't play rhythm with the rest of the team! It really will drive people crazy.