As a designer, is it a problem to use too many different fonts?

too many fonts are breaking the coherent of the design, just if you are making a funky style designs and the majority not for readability 

No at at all its not a problem, but you should know what kind a font represent your design Turly and nicely, don't use don't only because it look good every font got its meaning ...

Limit your fonts. A big part of putting together a good design, as you’ll see, is making sure the over-all look is consistent. The best way to accomplish a consistent look to your design is limiting the number of artistic motifs (themes) that you use. The fonts you select are the first variable you want to limit. I typically like to pick just2 fonts per design.

You should assort a collection of the fonts that you see are suitable to your design style and your projects also. And once you feel content about this collection, focus on leveraging your sense of typography to make the best out of these fonts in your designs.

Use only  two or three font per creation. You can use Font Explorer or SuitCase to activate and sort your font.

It depends on projects you designs it is ok for typography that will designed to T-shirts, posters depends on mode and theme, but for corporate identity and branding must use max of4 fonts and use professional fonts and fonts that never exist on other advertisements so it may look attractive elegant, and catchy.