Battery Ampere-hour rating vs Battery Amps (not an experienced user)
Your hunch that batteries have a current limitation is correct.  In general, it's hard to tell the current rating [A] from capacity [A&middot;h].  You have to look it up in the datasheet.  A lot depends on the design of the battery.  

For example: coin cells with 500mAh capacity may have only 3mA max current.
Another (opposite) example: automotive starter battery with 40Ah capacity may have 500A max current.

Lead-acid batteries are interesting in this respect, because there are two distinct types.


Starter lead-acid batteries are designed specifically to deliver high peak current for a short period of time.  Deep discharge, however, dramatically shortens the life of a starter battery.  So, it's not suited for routine operation at high depths of discharge.  Your typical starter battery in the automobile works at very shallow depth of discharge.
Deep cycle lead-acid batteries are designed (as name suggests) to discharge further.  But they can not provide as much instantaneous current.
Here's an example datasheet for a deep cycle battery. Have a look at the nominal capacity on p.1.  Notice that capacity depends on discharge current (i.e. the rate of discharge).


-

Depth of Discharge    Starter Battery    Deep-cycle Battery

100%                  12–15 cycles       150–200 cycles

50%                   100–120 cycles     400–500 cycles

30%                   130–150 cycles     1,000 and more cycles


(Source.)

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