The hypersensitive colleague can cause you trouble. How? Well, let's say you joke around with hypersensitive colleague about [insert anything, they are very sensitive after all]. Hypersensitive colleague complains to a partner/or management. Now, the supervisors may very well know hypersensitive colleague complains about everything; but, you might nevertheless be implicated in making a reportedly insensitive or even inappropriate comment.
You might be asking: this is a minefield...how will I know when I am stepping on a bomb? The problem is you won't know necessarily. But, if you hear someone could fall in this category, be very careful about jokes, misc. observations (especially of the personal type, etc.). Keep it very professional. Be overly cautious in your communications with this type of colleague.
I've seen some good questions in the recent comments posted here and will try to address some of them.
Have you done something to advance your career in the last week? Reached out to someone new? Started to learn about a new area?