...I haven't seen a single Bible verse that says that.
I think I recall Maureen O'Hara telling Henry Fonda that their preacher, Wally Cox (of Underdog fame), told her the Bible says, "As ye are, so shall ye remain", meaning the rich will always be rich and the poor will never be able to better themselves.
Can ANYONE find a verse in the Bible that backs up old Underdog?
These Scriptures below from 1 Corinthinans 7:17-24 come to mind, but it uncertain whether they are to what the characters were referring in the film. It is highly doubtful that the passage means that a person born into poverty is destined to live in poverty as his or her lot in life.
"But as God hath distributed to every man, as the Lord hath called every one, so let him walk. And so ordain I in all churches. Is any man called being circumcised? let him not become uncircumcised. Is any called in uncircumcision? let him not be circumcised. Circumcision is nothing, and uncircumcision is nothing, but the keeping of the commandments of God. Let every man abide in the same calling wherein he was called. Art thou called being a servant? care not for it: but if thou mayest be made free, use it rather. For he that is called in the Lord, being a servant, is the Lord's freeman: likewise also he that is called, being free, is Christ's servant. Ye are bought with a price; be not ye the servants of men. Brethren, let every man, wherein he is called, therein abide with God."