Often the strings will get caught in the nut. A properly cut nut will go a long way toward preventing this. Have a qualified repair person check yours. Rubbing a pencil over it to get some graphite dust in the slots can help lubricate it. Graphite and composite material nuts can work fine but a well cut bone nut is your best bet.
Whammy bars bring a whole new set of tuning problems into the picture.
If flawless tuning at all times is required and, at the same time divebombing trem work is part of your style, a good double-locking tremolo system is necessary. Systems that use a string lock behind the original nut may not prevent catching problems. Even a double locking system requires proper maintenance and new strings for accurate tuning. If you need to wham, but don't want to deal with a locking system there are other alternatives and some tricks for staying in tune. Proper nut lubing helps, too, as does a trem block and saddles that allow the string to slide smoothly.