Recently started having this problem with my Rebel. It's about 4 years old.
If it is in AF mode, I will put the button down halfway (like always) to focus, but it will not take an actual picture. It just clicks through and doesn't do anything. Happens about 90% of the time, I'd say.
It works fine with MF. Has anyone had this problem? I briefly googled it but didn't see much.