There are no *primitive* solutions, even when one of the fourth powers is replaced by a square. This was proved by Bennett, Ellenberg, Ng (see Int. J. Number Theory 6 (2010), 311-338). 

Non-primitive solutions there are plenty, see Noam Elkies's comment for an example.

P.S. I found the paper online [here][1], I don't know how long the link lasts.


  [1]: https://pdfs.semanticscholar.org/9bd7/a3aae8b6b368c847035addd0140af12f4bda.pdf