Species from the same genus can usually interbreed. Sometimes species from two closely related genuses (geni?) can breed. When you get further up the tree than that, there's a lot less chance that it could work. There are quite a few hybrids between different macaw species. There are also a lot of hybrids between different conure species.
Cockatiels are the only species in their genus, which is Nymphicus. A cockatiel has successfully produced babies with a galah, which is a cockatoo from the genus Eolophus. There are other cockatoo species that can breed with each other, but I don't know whether all of them could do it. They probably couldn't breed with a bird that wasn't a cockatoo. According to this link, there have been some hybrids between galahs and other cockatoo species that occurred in the wild:
https://zoologica.wordpress.com/tag/hybrid-cockatoos/Budgies are also the only species in their genus, which is Melopsittacus. There are some old claims about budgies producing young with lovebirds and canaries, but the claims haven't been verified and aren't believable. As far as I know, there are no verified cases of a budgie hybridizing with anything.