Some require cookies to be enabled (Facebook I know does), so they networks won't let you log in in a private mode.
I find it a little annoying though that cookies are not sandboxed by tab, rather than by window/session. If I log into FB in incognito, and then do a little more private browsing in a new tab the FB cookies are still accessible in the other tab.
I guess I could be more vigilant with my browsing habits but I think this is a fair feature to implement at browser rather than forcing user to jump through more hoops to protect privacy. On a side note, when will chrome finally offer API hooks to allow NoScript to be developed for it?!