You start out with 5 spots available for your bunnies and as you level up in Tunnel Town, you will get more bunny spots. You will get a total 30 bunny spots for your burrow. However, you can put away a bunny anytime by tapping a bunny to open up their bunny card, then tapping the "put away" button. This will put your bunny into storage. You can take your bunny or bunnies out of storage any time by tapping the icon on the right side with two bunnies on it and dragging the bunny you'd like to use into your burrow. Since the bunny storage space is unlimited, you don't have to give up any of your special bunnies, you can just put them away to use later.
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