YES, anger, as well as sadness, CAN build up. This is a well known phenomenon. It is common for one to have these emotions build up from repeated and large cases of problems, to the point where even the tiniest of annoyances they normally wouldn't sweat will finally make them burst into a rage or tears.
This has happened to me plenty of times, too. I very rarely cry, but once at school, I was so stressed out by happenings then that when I zoned out and someone sternly told me "sit down, you're out" when I had been chosen out during a classroom game, I couldn't stop myself from sobbing at my desk. All the attention I got from that only made it worse.