For anyone reading this who doesn't memorise error numbers.

Error 833: SQL Server has encountered <n> occurrence(s) of I/O requests taking longer than 15 seconds to complete on file [<full path of file>] in database [<database name.>] (<dbid>). The OS file handle is <file handle>. The offset of the latest long I/O is: <offset in hexadecimal>

Yes, it's an IO latency problem. IO requests, which should take no more than 50ms are taking over 15 seconds.