power rating of supercap
For more conventional capacitors, see this answer on SE. There, you will find a link to 'Selection of Capacitors for Pulse Applications', relating to Polypropylene capacitors. It is not limited entirely by thermal dissipation.

This same company (WIMA) offers supercaps for power applications, like cars and wind storage, and they do have current ratings. Here is the WIMA supercap_r page, where the lowest current rating, for a 100F supercap, is 30Amps (r=12mOhm, so 10 Watts). A note on the page implies that current ratings for their supercaps are strongly related to thermal dissipation: "Due to the large case surface of the cells heat being generated by the continuous current flow can better be drawn off."

An attempt to get some sort of metric from their data gives me from 120mW/gram to 250mW/gram for their 90 to 120 gram capacitors. 

The little CooperIndustries 3F supercap is 8grams, so (using 100mW/gram) a limit of 800mW is predicted, or 4 Amps.

All that could be worthless info, however. The little cap may not have been designed for anything but a backup battery replacement, with no attention to the details needed for high continuous current use.