The concept that God will not put more on us than we can bear does have some biblical support as long as one keeps in mind the needed balance between what God sovereignly allows according to His wisdom and purpose(s)and our human responsibility to trust and draw near to Him. For instance, compare 1 Corinthians 10:13, 2 Peter 2:9; Proverbs 3:5f; Psalm 37:3-6. We have the responsibility to cast our burdens on the Lord and He promises to not allow us to be shaken (cf. Psalm 55:22 with 37:23-24). In Psalm 68:19, God is called our “burden bearer.” See also Psalm 81:6-7 and Matthew 11:20-30. This is further supported by 1 Peter 5:6-7. Casting our cares or burdens on Him is presented in this text as an incentive for humbling ourselves under the sovereign purposes and work of God.
You might want to take a look at the lesson, Counsel Concerning Our Cares, which is a study of this 1 Peter 5:6-7 passage on our web site. The basic principle here is that—as our problems are, so shall the Lord be. This means we will experience His sufficiency in our trial(s) if we will rest in Him as delineated in Scripture.