You can definitely do it for under $20,000 for the basic 518 cards, but you will have to settle for some very poor cards along the way, and you won't be able to get any rare backs. You will pay at least $1,000 each for Demmitt and O'hara St. Louis variations, which takes you to 520 cards. The big 4 (which includes the original Big 3 plus the Doyle NY Nat'l) would be impossible on the budget you describe.
But I wish you good luck. I have really enjoyed putting together my collection.
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