In science-related graduate schools, it is quite often the case that students will not be accepted into the program *unless* the department has enough funds to offer them an assistantship of some sort.  Students who try to do it all *on their own* often find themselves under even more pressure than a funded student.  On top of trying to pass extremely difficult courses and pursue original, cutting edge research, they may find themselves also working at least one other job that barely pays rent, much less tuition and all other debts incurred along the way.  Often, unfunded students succumbs to financial pressures and drop out to pursue more financially stable opportunities.  

Students dropping out of graduate programs also make their host departments' statistics ***look bad*** in the eyes of their superiors (i.e. deans, university president, provosts, etc..) and can lead to diminished support for those graduate programs.  Since universities don't want to hurt their own reputations (or lose state/donated funding), they tend to be selective of their graduate students.  And I believe this is a major reason why self-funded students are often not even allowed in graduate programs:  statistically speaking, their success rate is likely too low to merit taking a chance.

Of course there are exceptions (e.g. having wealthy parents or pursuing non-science graduate programs), but it is certainly a **red flag** if a student willingly tries to pursue a graduate degree in the sciences without any source of funding.