We adopted a 3 1/2 year old cat from a no kill shelter. I was able to meet the previous owner (whom had to get rid of her due to health reasons). Our cat Mystie, was litter box trained with no accidents and now she is having accidents all over. Could she be doing this due to a new environment? Is there anything I can do to help her>?

By accidents do you mean she is urinating or defecating outside the litter box and has she been doing it since she has been at your house or were things normal in the beginning?
A urinary tract infection can make a cat seek out other places to go as they associate the discomfort of the infection with the box. It may be wise to get her in to see the vet to rule out any medical issues.

Maybe you did this already, but if not, and if you can contact the previous owner again, find out what brand of litter he used, and make sure you are continuing to use the same kind. Many times the smell and texture of a particular brand becomes associated with the process.