When should you split a database into 2-3 databases for performance reasons?
Size is not really a reason to do a database split.  If there is slow query performance or other factors like that, then maybe.  But keep in mind, splitting a database using the same hardware and storage will provide very few gains.  There is also the possibility of partitioning the data which will accomplish something similar.

The problem is that this was devised as a solution that it may not fit.  I have multi-terabyte databases that respond faster than 100 GB databases because it is tuned and indexed within an inch of it's life.  Replacing hardware will often cover up poor database and query design.  An immediate fix would be to add more RAM.  8 GB would be the minimum I would want to see in a SQL server, but given the size of the database I would suggest a minimum of 128 GB.

Ultimately we need more information on what performance issues you are experiencing in order to guide you to a more complete solution.