There are certain musical qualities that the average listener has a better feeling for than the semi-educated classical musician. It didn't always use to be this way and I don't like to make a bourgeoise distinction. But unfortunately, with the deterioration of classical music as the popular form of entertainment and the emergence of new forms of music, the result has been a loss of the physical feeling of music- the groove. Maybe we can call it the over-academisation of music.  Today there is a horrible lack of feeling among some well-trained classical musicians at achieving a basic rhythm that you can bounce your head to. (I have conducted no studies. I just speak from experience.)
Hip-hop musicians get it. Feeling a groove that the listener can sing along to comfortably is of utmost importance right from the start. If it doesn't have bounce, it's not good enough to inspire a rap. But of course classical music doesn't always asks for this kind of constant, stable groove. The rubato Arthur Rubinstein played Chopin Mazurkas with is a kind of flexible sense of rhythm that is on a higher plane than the topic at hand now (though related). I want to talk about a more basic feeling. I'm talking about the kind of steady rhythm in Bach's Brandenburg Concertos that makes you want to tap your foot to. The kind of whimsical rhythm that the best performances of Le Nozze di Figaro's Act II finale make you skip to with glee and forget about whatever superficial high you were contemplating earlier. The kind of rhythm to which Mendelssohn's "Scottish" Symphony makes you want to put on a long skirt and jump joyously around the kitchen. Rhythm is a basic quality that humans have. My point is that its emphasis is severely lacking in classical music education.
As classical musicians, we are all taught to stay on top of the beat. If you don't anticipate, you can get behind or be late. This is for precision and more so for ensemble playing. No one likes a sloppy group. The danger, however, is becoming too mechanical, less natural. When we're playing in a groove, and to really feel that groove, we need to sit back a little. Some conductors hesitate calling it "playing behind the beat", but I really don't think they should. Something needs to be said to make you chill out, or else you'll totally miss the affect behind the music. I doubt Mozart wanted a tight-ass playing Figaro.
I remember playing Son of Chamber Symphony by John Adams at Mannes and the fun we had when we were locked into the rhythm, bouncing along like Jay-Z at Madison Square Garden. It felt so cool and natural. But when the tempo changed abruptly during rehearsals, usually someone pushing foward too much, that groove was lost right away. I just do not believe that enough emphasis, in the university setting, is placed on rhythm. It's hard to teach, I know. Pitch, matching your tone, phrasing, are more tangible to explain. But rhythm, I believe, only gets better from experience. When the feeling is right, an audience knows it. They smile. They feel good. Getting this right is so important, that seriously maybe we should all start listening to more hip-hop. Lean back. Go'on brush your shoulders off.