As markysngc rightly said, in general most of the program files are removed, however often a program will leave some folders and other data files, which are not always removed by the add/remove utility.
vehelpfulguy is also correct in what he says if we are talking about data files which the user has created, and yes if they are sensitive files i.e. financial etc, then yes you should be aware that anyone with a little knowledge could access your files even after you have 'deleted' them.
Your question however was about removing programs not data files, so the answer is yes, most of the program files will be deleted, but as I said sometimes there can still be .dll files or more importantly, entries in the registry which could cause your PC to run slowly or even crash if not removed.
After using add/remove, I then locate the original folder where the program was stored, and I then delete all files and folders permanently i.e. not to the recycle bin. To clean up any problems in the registry I then use a registry cleaner to locate and delete any possible orphans.