Does the Bible say that God won't put more on you than you can bear?

No it doesn't. Here's what it really says:

1 Corinthians 10:13 KJV "There hath no temptation taken you but such as is common to man: but God is faithful, who will not suffer you to be tempted above that ye are able; but will with the temptation also make a way to escape, that ye may be able to bear it."

NKJV "No temptation has overtaken you except such as is common to man; but God is faithful, who will not allow you to be tempted beyond what you are able, but with the temptation will also make the way of escape, that you may be able to bear it."

NIB "No temptation[a] has overtaken you except what is common to mankind. And God is faithful; he will not let you be tempted[b] beyond what you can bear. But when you are tempted,[c] he will also provide a way out so that you can endure it."

HCSB "No temptation has overtaken you except what is common to humanity. God is faithful, and He will not allow you to be tempted beyond what you are able, but with the temptation He will also provide a way of escape so that you are able to bear it."

AMP "For no temptation (no trial regarded as enticing to sin), [no matter how it comes or where it leads] has overtaken you and laid hold on you that is not common to man [that is, no temptation or trial has come to you that is beyond human resistance and that is not [a]adjusted and [b]adapted and belonging to human experience, and such as man can bear]. But God is faithful [to His Word and to His compassionate nature], and He [can be trusted] not to let you be tempted and tried and assayed beyond your ability and strength of resistance and power to endure, but with the temptation He will [always] also provide the way out (the means of escape to [c]a landing place), that you may be capable and strong and powerful to bear up under it patiently.

The Message "No test or temptation that comes your way is beyond the course of what others have had to face. All you need to remember is that God will never let you down; he’ll never let you be pushed past your limit; he’ll always be there to help you come through it."