Can a cockatoo mate with a cockatiel?

Generally, no - parrot species cannot interbreed, with the exception of macaws in the Arinae sub-family.

However, macaws are neither cockatoos, nor cockatiels, as the question asks.

There have been instances of cockatiels cross breeding with galahs, which are small members of the cockatoo family. If the question refers to the cross-breeding of cockatiels and the sulphur-crested cockatoo, commonly kept as a pet, then to date, no such hybrid has been produced.

See the related link below for an article on the dubious practice of breeding hybrid parrots.