As far as social issues go, I don't think people are being too sensitive. IMO people saying racist/sexist/homophobic/etc. stuff is unacceptable (as well as making fun of someone for a traumatic incident, a disability, etc.) but a lot of people try to pass it off as being "edgy" or a joke when in reality they're just being a jerk. I don't think you're being too sensitive if you're offended by, you know, things that are offensive.
BUT. I do think when someone says "Hey, I think this movie sucks" and then they have 30 people in their Twitter mentions angry at them....those people are being too sensitive lol. When it comes to a difference in opinions on movies, TV, music, celebrities, etc. I don't think people should get offended so easily. It's not that deep! Unless the statement is something like "This movie sucked because the main character was gay" "This movie sucked because it had too many women in it" then it shouldn't be something people get too wound up about. "This movie sucked because the plot made no sense and the acting was crappy" isn't offensive IMO lol.
But lately when people talk about others being "too sensitive" or "too quick to be offended," they usually mean about social issues so.....