Take, for example, the life of Jeremiah. He is remembered as "the Weeping Prophet". He served the Lord faithfully. He did exactly as God commanded him. He preached the Word so hard for so long; but no one listened. No one repented. Jeremiah was despised and rejected, even by his own family.
In the end, the disobedient people of Israel were handed over by God as captives to Babylon. Jeremiah was neither honored nor influential among his peers. But he was successful in doing just what God called him to do.
Jesus Christ succeeded too, by leaving His glorious throne to become a lowly, poor, human being. He was despised and rejected, and submitted Himself to death by crucifixion (Isaiah 53: 1-10, Philippians 2:5-8).
All the apostle, as well, faithfully served Christ, working hard to spread His gospel. And for their service, they all endured hunger, mockery, imprisonment, floggings, beatings, and stonings. All despised and rejected, only one of them, John the Revelator, survived to die of old age, a prisoner in a Roman rock-yard.
So, what does God promise? I'll get to that in a moment.
The American Dream promises health and wealth and worldly success to anyone willing to work for it. But the American Dream can't keep that promise toward everyone who works hard because they're all competing against each other. Competition is the ongoing quest to defeat one's fellow human beings. That's not how the kingdom of God works, but that is how the world works. Therefore, someone has to be defeated, (according to the world's standards of success and defeat) even if they tried really, really hard, even (perhaps especially) if they faithfully serve the Lord. And in a competitive world it's difficult enough for everyone just to get job, let alone forge an actual career in a choice field. Some will succeed. Many will not; even though they do their best.
God does not validate the promises of the American Dream. But what He does promise is that He will be with you through all the joys and inevitable sorrows you encounter as you serve Him. He promises every spiritual blessing of spiritual comfort, joy, and peace in the midst of tribulation (Ephesians 1:3, Philippians 4:7, John 14:27, 1 Peter 1:6-9). And your reward for all your service will be in heaven (Romans 8:18). And that's what God calls "success".