The song is very catchy. It's also really wrong. The passage of Scripture that people tend to appeal to for this idea is usually 1 Corinthians 10:13b "God is faithful; he will not let you be tempted beyond what you can bear." It's important, however, to look at context. No verse is an island and so we have to look at what is actually being said in the verses around that phrase to figure out what it actually means.
1 Corinthians 10 is talking about the example provided by the Israelites and their history. They had Moses leading them, they saw God provide escape through the Red Sea and yet almost none of that original generation made it into the promise land. They all fell to sin. Paul then tells the Corinthians that this was a warning provided to them not to fall into sin themselves. The temptation to sin is prevalent in the life of every believer and the Corinthians needed to be aware. They had hope, however, because he includes the promise that they would not be tempted beyond what they could bear. The following verse clears up any misunderstanding "he will also provide a way out so you can endure it."
This passage was clearly talking about how God will not let us be tempted into sin beyond what we can bear. As believers we never have to sin. (Romans 6) God promises that since we are dead to sin that He will always provide an opportunity to not sin in a situation. We don't have the ability to claim that "the devil made me do it." We won't always be successful but that's why we have the Holy Spirit who aids us in our spiritual growth (sanctification). This passage does not apply, then, to the idea that hardships in life will not be overbearing. I don't think that idea rings true to our actual life experience and it gives the idea that either God has misidentified our strength or that we are failures because we are not going through life with a constant smile.
Why does this difference in understanding 1 Corinthians 10 matter? It matters because the idea that "God will never give us more difficulties than we can handle" only promotes the idea that we are capable in and of ourselves to handle whatever situation we're struggling through. The truth of the matter is that God often gives people more than they can handle because it's in that struggle that we realize it is only in Christ that we can find refuge and strength to endure.
We can see this in Scripture. Recently, I've spent time in Psalm 13, a passage that demonstrates a little more clearly what life was like for David when he was given hardship beyond what he could deal with:
How long, LORD? Will you forget me forever?
How long will you hide your face from me?
How long must I wrestle with my thoughts
and day after day have sorrow in my heart?
How long will my enemy triumph over me?
Look on me and answer, LORD my God.
Give light to my eyes, or I will sleep in death,
and my enemy will say, "I have overcome him,"
and my foes will rejoice when I fall.
But I trust in your unfailing love;
my heart rejoices in your salvation.
I will sing the LORD's praise,
for he has been good to me.
David clearly seemed at the end of himself. It is at the end of ourselves, though, that we see that God is the one who delivers. Sometimes His Salvation comes in the form of deliverance from the difficulty and suffering that we're facing. Other times, we are kept in that suffering because even though we may never understand it, our suffering brings God more glory. What we can count on is a God who loves us and who desires for us to grow in our sanctification and our love for Him. Instead of clinging to the lyrics that He "might let you bend, but he won't let you break", it might be better if we sought out these lyrics instead:
Brokenness, brokenness is what I long for
Brokenness is what I need
Brokenness is what you want from me
So take my heart and form it
Take my mind and transform it
Take my will and conform it
To yours, to yours, oh Lord.