<p>If I add 3-400 databases to a single SQL Server instance will I encounter scaling issues introduced by the large number of databases?</p>

This is one of those questions best answered by: Why are you trying to do this in the first place? What is the concurrency against those databases? Are you generating databases when you could have normalized tables to do the same functionality?
That said, yes MSSQL 2005 will handle that level of database per installation. It will more or less be what you are doing with the databases which will seriously impede your performance (incoming connections, CPU usage, etc.)