PostgreSQL WAL recovery not working due to missing WAL
I've been kicking PostgreSQL backup around the last couple of weeks, because of a new job, and I find the implementation of PostgreSQL's backup quite hard to grasp.

What I have learned however, is that the a missing WAL file is the worst situation you can have. A missing WAL file will not allow you to continue your restore past the missing file, because the PostgreSQL database relies on the WAL file to be ACID.

You could try to retrieve some of the information from the non-restored WAL files, but you wouldn't have any guarantee, that the resulting database/instance is consistent.