What is the typical DPI at which playing cards are printed?
From a quick Google search it would appear the standard industry DPI is 300. Another web printing service discusses is here: A Note About DPI. They only print at 300 DPI.

As is mentioned in a comment Graphic Design may be more informative: https://graphicdesign.stackexchange.com/questions/95/what-dpi-should-be-used-for-what-situation

As to why your cards came out incorrectly there could be any number of issues including incorrect exporting or due to compression their end*. You should always check their FAQ in future:


  To customize your poker cards, create images that are 300DPI and CMYK color mode. Allow bleeding area as specified in our product page. To ensure the highest quality of printing, please use high resolution images.


In my experience with any kind of printing it is hard to get things perfect first time. There are a lot of tools out there (e.g. Publisher can be switched to CYMK mode) but just checking again and again and following the manufacturer guidelines seems to be the best way to improve your results.

*I wouldn't be surprised if they converted SVG to another format incorrectly.