Sorry. I hear 'thick' for fat or stupid pretty regularly, and I'm in the US. More importantly,
definition 2: something resembling a thicket in density or impenetrability, a tangle
... usually the context is that a thicket is a daunting obstacle to reaching an understanding or a goal. It has a pretty strong negative connotation, such that I'd expect a person nicknamed Thicket to have very messy hair, or else to be unpleasantly prickly and not very bright. As a given name, I just think it's awful... it's awful.
Wait a minute ... are you joking? Thicket
Katherine? This honestly strikes me as funny, it's so bad.