We had this problem after we took a database from PROD and restored it in QA. Our application called three databases on three servers and aside from that was not complicated.
It turns out a spurious SPN (Service Principal Name) was getting in the way of the service account under which the connection should have been running. We discovered this using the Program Files > Microsoft Kerberos Config Manager.
The short term fix was to use SQL Server Configuration Manager and change the SQL Server and SQL Server Agent connections from the service account to 'LocalSystem' under 'Use BuiltIn Account'.
