Can output current be higher than what the input source can provide
The rating of the battery is probably Ampere-hours, not Amperes.

The Ampere-hour rating of a battery relates to the amount of energy stored in the battery, not to the current it can deliver.  A 2 Ah battery could theoretically deliver 2 Amps for 1 hour, or 1 Amp for 2 hours, etc. If you draw 4 Amps from a 2 Ah battery, it will be fully discharged in less than 30 minutes (If you discharge a battery at a high rate, you will get fewer Ah from it than the rating indicates)

After further thought, I noticed that you said the battery was 5 volts - as there is no battery chemistry that will directly produce 5 volts, I suspect you have a "power pack" consisting of a battery and a DC/DC converter, designed to charge cell phones, rather than a plain battery.  

In that case, the 2 Amp rating would be the designed output rating of the DC/DC converter, and attempting to draw more than 2 Amps from the power pack may damage the DC/DC converter.