That is nonsense. Everyone has limited bandwidth for a fact. Be it 256Kb/s or 20Mb/s or whatever.
When an application starts to upload or download something and takes all your bandwidth all other activites which are depending on that bandwith come to a halt. That is certainly most visible in the case of upload as the upload bandwidth is usually lower than download.
That is the reason why applications/services which are usually used in the background (as in dropbox, ftp, torrenting clients etc) all have throttling controls so they don't take all your bandwith so you can still stream internet radio, browse internet at acceptable speed etc.