Multiple times in the Gospels Jesus says something along the lines of "Whatever you ask for in my name you will receive" (Matt 7, John 14, etc). So, we ask. We pray believing that we ask as God's representatives in the situation. We ask in his name… and nothing changes. We remember Jesus' parable of the persistent widow and we continue to pray and still nothing changes. I have prayers I've prayed for years and still nothing changed.
When reality confronts our hope there is often a divide between the two. We don't know what to make of the Bible's promises regarding prayer. The temptation is one of two things:
1) We become bitter at God.
2) We stop hoping for the big things and pray prayers that are more in line with reality. Instead of praying for healing we pray that the doctors would have wisdom.
There probably is a 3rd option here as well:
3) one in which people ignore the reality of their situations. People claim to be healed and still die of cancer. The pain and frustration involved with living in the tension of unanswered prayer becomes too much.
It's hard to live in the desert between God's promises and the pain of our situations. Life is hard enough and then we add the pain of testing our faith. All sorts of questions come to mind when God doesn't answer prayer; the scariest of all is the one calling into question the veracity of what we believe. If God says he will answer our prayers and my prayers don't get answered then maybe it's all a hoax. All of a sudden faith seems like a house of cards about to collapse.
I think this is where experience comes in. Our history with Christ becomes an anchor for the soul. In our confusion and desperation our minds cry out for something to grab hold of and the reality of our experience provides a footing. That happened. God revealed himself. He spoke to me. This is real. And from that ledge we catch our breath and re-assert our faith--I don't know why this has (or hasn't) happened but I do know God is good and He is in control; we remind ourselves of what is true and continue leaning on the nature of our Heavenly Father.