On Kindle Fire, the Marketplace is happy to take my money, but not let me download what I paid for.
Relogging does not fix the problem.
Restarting does not fix the problem.
Reinstalling does not fix the problem.
Please fix the problem.
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We're tracking this issue at MCPE-54531, so I will resolve and link this ticket as a duplicate. You can try clearing the Marketplace cache to see if that helps.
Navigate to Settings > Storage
Press the Clear Marketplace Cache button at the top
Exit Minecraft and restart your device