Your hunch that batteries have a current limitation is correct.  In general, it's hard to tell the current rating [A] from capacity [A·h].  A lot depends on the design of the batteries.  Coin cells with 500mAh capacity may have only 3mA max current.  Lead-acid batteries are interesting in this respect.  

[more coming up]

**p.s.**  If you want to read-up, here's an excellent web site on batteries - [Battery University][1].


  [1]: http://batteryuniversity.com