I used to say those phrases. A lot. It helped me cope as I was struggling under the weight of disappointments, health issues, you name it. I even had some folks speak those words to me most recently as I went into the hospital for surgery - "Remember! The Lord won't give you more than you can handle!"
One day it occurred to me - in all my years of reading the Bible, I've never seen that verse. Anywhere. So a few years ago I did some searching and I made a startling discovery: the Lord, in all of Scripture, has never said that He won't give us more than we can handle.
As I kept digging, I discovered it was Mother Theresa who wrongly started that idea when she said, "I know God won't give me anything I can't handle. I just wish he didn't trust me so much." I realize that saying, "Mother Theresa was wrong," is nigh to heresy, so I say that with love, respect, and truthfulness.
All throughout Scripture people were handed impossible tasks - Moses freeing thousands of Israelites, David slaying a giant, Joseph sold into slavery and becoming the most powerful man in Egypt, Daniel in the lion's den, Mary becoming a pregnant virgin, etc. Do you think any of these "heroes" of the Bible were in the middle of such extremely stressful circumstances and thought, "Well, God won't give me more than I handle, so I'll just keep hanging on"? The point was to show that a possible God is in charge of impossible situations. In all of these stories, the main character consistently depended upon God for deliverance, not themselves.
The Apostle Paul wrote a letter to the Corinthians saying, "We think you ought to know, dear brothers and sisters, about the trouble we went through in the province of Asia. We were crushed and overwhelmed beyond our ability to endure, and we thought we would never live through it. In fact, we expected to die. But as a result, we stopped relying on ourselves and learned to rely only on God, who raises the dead. And he did rescue us from mortal danger, and he will rescue us again. We have placed our confidence in him, and he will continue to rescue us" (2 Corinthians 1:8-10, NLT, bold/italics mine).
The pressure of the problems on Paul was so great, he thought he was going to die. But the feelings of being crushed by his problems stopped once Paul stopped trying to handle it ("we stopped relying on ourselves") and gave it to God ("learned to rely only on God, who raises the dead"), then God came to the rescue.
What God has said is, "When they call on me, I will answer; I will be with them in trouble. I will rescue and honor them." (Psalm 91:15)
Jesus says, "Come to me, all of you who are weary and carry heavy burdens, and I will give you rest." (Matthew 11:28)
The biggest lesson that keeps being reinforced with me this year is that I am not meant to handle my problems. If the Apostle Paul couldn't handle it, why should I??
I need to continually give these stresses/anxieties/problems over to God. I need to rely on God completely. I am not meant to handle these problems, only God is. The only way to have rest is to give God my burdens and wait to see how He is going to work it out. When I'm feeling buried under the weight of my problems and it's more than I can handle, it's a sure sign that I'm trying to carry the load instead doing what I'm supposed to be doing: giving it to God.
I hope this gives you some freedom today knowing that you are not meant to carry the things that are weighing you down. Remember: God won't give you anything HE can't handle...and He promises that His burden is light (Matthew 11:30).