There are no limitations to number of data sources (we have servers with thousands here at Tableau). The use of a high number of data sources may result in slow behavior based on the size of the data, the type (live vs extract), the joins or cross data source joins, etc..
If you are experiencing issues with performance and have followed suggestions for designing dashboards for the best performance - 6 Tips to Increase Dashboard Performance - then please reach out to Tableau Support for additional assistance.
Patrick - The OP suggests that they have a server core based license. In that case your assertion is accurate.
The OP may be mixing this with the Tableau for Embedded applications, which is a different licensing model. In that scenario - there will be a limit to the number of data connectors used. That is a licensing limit as opposed to a performance constraint. I'd also like to distinguish data connectors, from data sources. You can still have as many data sources as you need, but the Embedded Analytics Pricing for Tableau limits the number of data connectors used.
Stephen - I hope this clears up any confusion. If there are still questions, I'd recommend connecting with your respective Tableau Sales Account Manager to clarify.