My MP3s split up

[roggerdog1974]

I have hundreds of mp3 files and ALL play perfectly on my Dell computer; however, when I play them on a "stand-alone" player using a CD, about 10% of the songs will be split up into several parts. The songs that are split will go from part 1 to part 2 to part 3 ... just like when a song has ended and another begins. Some of the problem files are ones that I've downloaded off the net and others are ones that I've ripped myself. It seems to be random on which ones will be divided up. I've used 3 different stand-alone units, 2 DVD players with mp3 capability and a personal CD player with an mp3 option, they all respond the same. I want to get an IPOD but I want to get this problem resolved first. Once again, ALL the songs play perfect on my computer using either Winamp or Windows Media Player. The one thing I don't know is if all 3 stand-alone units have split up the problem files exactly the same using the same CDs. Oh yes, I've tried several different CDs packed with mp3 files.
I know there are plenty of people out there who are smarter than me so I'd really appreciate any help I can get.
Thanks a bunch,
Dave