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.

The original, unpleasant meaning referred to people, usually wounded soldiers, who were quadruple amputees (no arms or legs). From there, it came to mean "in a hopeless situation and unable to do anything at all." "It was totally crazy at work. Everything was frantic and disorganized. I'm exhausted. I'm completely wiped out. I'm a basket case. I don't have the energy to do anything but sit and watch TV."

"Predicament" is a fairly formal word. It means that you are trapped in a difficult situation and don't see a good way out. "I'm in a predicament. I haven't logged into this account for years. It doesn't recognize my computer and says I can't log in until I enter a validation code. It sent the validation code to an old email address I don't have access to any more. I can't update my email address in this account, because I can't log in." Here are two idioms referring to a similar situation. Don't try to use them, but understand them: a) "I'm on the horns of a dilemma." (Rather formal). b) "I'm caught between a rock and a hard place. (Informal and somewhat regional).