SQL Server Mirroring (with Witness) and Client Timeouts
When to failover is decided by the Witness server not but the Primary or Secondary servers in a mirroring session. 

The Secondary Server will come online if the Witness Server cannot see the Primary Server. The Secondary server only gets transactions from the Primary server and nothing else. If the Primary Server stops sending Transactions the secondary server will just simply sit there in restoring mode and do nothing, But if Primary servers becomes invisible for Witness Server thats when the failover happens and the Witness server will make Secondary server to come online. 

Actually this the whole purpose of having a Witness server(Auto-Failover). If the secondary server could make these decisions on it own we would not need a witness server at all :) .