A predicament is when you are in a situation where you cannot figure out what to do because you have to choose between several things and the results may be negative. For instance, if you only had enough money to pay your rent or to buy food, that would be a predicament. If you do one, you suffer the consequences of not having done the other.
A "Basket case" is someone that is troubled and has a lot of problems in their lives such that they tend to not be very useful to have around. For instance, if a co-worker were to get a divorce and then become very depressed, they might start making lots of mistakes. Your co-workers might then say that the person has been a basket case since the divorce.