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It's a two part answer

1. Enough hands to prove you are a winner are limit X before going to X+1

2. Enough hands to build an adequate bankroll for X+1.

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I figured as much, but how many hands is enough? 5k? 10k? 100k?

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Start phasing in a new limit as you approach 20k hands, depending on how well you're running. If you multitable, try reducing the number of limit X tables and add one table of limit X+1. Do this for part of one session, then switch back to normal. The goal is to slowly build a database of hands at X+1 before shifting your entire play over there. If you don't multitable, do you by taking one shot per n number of sessions. Slowly decrease n until you are ready for the new limit.

I would also advise against moving up limits too soon.