Each bird specie has different limits. Please see the table below:

Flippy Birds

You can only buy 2 Flippy birds, total. Once you've bought up to 2, you can no longer buy additional Flippy birds, at all.

Frosty Birds

You can only have 3 Frosty birds in your birdhouse at a time. When your Frosty birds die, you can buy more to replace them, but the most you can have in your birdhouse at any one time is 3.

Fruity Birds

You can only have 2 Fruity birds in your birdhouse at a time. When your Fruity birds die, you can buy more to replace them, but the most you can have in your birdhouse at any one time is 2.