Short answer: not really.
Depends on the school. Some have Nov-Dec-Jan 1 deadlines (especially some of the West Coast schools), but others' deadlines aren't until Jan 15th, Feb 1, Feb 15th, or March or April. For schools with rolling admissions, the earlier you submit your application, the better, but while there are probably some early birds hoping to beat the competition, I wouldn't call it "late" until after the deadline has passed. (If you're just starting your applications and haven't lined up letters of recommendation, GRE scores, or transcripts, then I'd say you're verging on late for schools with deadlines in the next month. Get a move on!!!)
Procrastinator that I am, I didn't submit anything until January 1st, I think, yet had no problems at any of the schools (none of which had earlier deadlines). I did the apps in order of deadlines, and submitted some as late as March, though if you're more efficient than I am, you might be able to complete them more quickly. If you are just starting to put your applications together, I'd recommend doing what it takes to make arrangements for all the supporting documentation (letters, GRE, transcripts) first, since those are the loose ends that can delay applications and tend to take longer to process. Check on these until you're satisfied they've been mailed and received! Then as long as your bits (statement, forms, resume/CV) are mailed a safe margin before the deadline, you should be fine.
Good luck, and happy New Year!