When I was looking through the Bible, for verses that speak about this, I came across a potential death-blow to the Calvinist teaching on irresistable grace (the "I" in TULIP):
Prov 1:24 Because I have called and you refused to listen, have stretched out my hand and no one has heeded,My eyes bugged out when I read it. I haven't done any extensive research on this passage yet, but it appears to explicitly reject the idea that when God calls you, you can't help but respond to it. I'm still waiting to see what the response will be.
If you know of any verses that either reject irresistable grace or affirm the reality of actual grace, please leave a comment and let me know. For info on irresistable grace, see A Tiptoe Through Tulip and scroll down to "Irresistable Grace." For info on actual grace, go here or here.