The message that "God only gives you what you can handle" makes me somewhat uncomfortable because it's said so often, yet it's... pretty much exactly the opposite of what the Bible teaches.
God doesn't test you to see what you're capable of, nor does he guarantee you'll be capable of everything you're put through. What the Bible does teach is that "I can do all things through Him who gives me strength." (Philippians 4:13) It also tells us that He uses the weak to shame the strong - the very fact that we're not capable in ourselves is used to His glory.
I think that often, the implications of that saying aren't fully realized. It can even be said by people who will quote the verse from Philippians with utmost sincerity. But it's dangerous, because God will give you tests that you can't handle, and we need to remember that all of our strength comes from Him. Trials aren't a test of merit, they are an opportunity to show the glory of God through your life.