No. You cannot keep chickadees as pets.
All native wild birds are protected in the US by federal law (Migratory Bird Treaty Act). No native birds can be kept as pets - by anyone.
You cannot possess a bird (alive or dead), nests, eggs or even feathers. It is also illegal to tamper with nests, or harass birds.
These laws are designed to prevent people from decimating the wildlife populations by poaching for illegal pet trade. Illegal pet trade has caused massive destruction to animals in other countries, some birds are threatened with extinction in their native countries.
The only persons who can handle or possess native birds, are those who hold state or federal licenses, like rehabbers who rehabilitate injured birds and re-release into the wild, biologists and researchers for specific studies and banding, wildlife educators, zoos and museums.
Please enjoy the wild birds in their native and wild environment - where they belong.
For more information regarding wildlife in the US:
http://www.tc.umn.edu/~devo0028/