Assuming the 15A was installed because the associated wiring is 14ga, then no, you cannot replace the 15A breaker with a 20A.

To be a little more specific, open up one of the outlets and check the color of wiring to them. If you have yellow colored sheathing then you have 20A wiring, or 12 guage. If it's white, then you have 14 guage. I would tend to agree with the above comment, the 15A breaker was used because the circuit was wired with 14 ga wire.

Color coding the sheathing of NM cables in fairly new. In most likely hood that cable does not have it.

Also it might have been wired with conduit or BX.

In any case the only reliable method is to check the wire guage.

But most cases if it was had a 15 amp breaker then the wiring is 14 guage.

Another problem with older wiring you never know what changes someone might have made. A circuit could have been extended with #12 wire because that is what he had handy, but that the orginal run was done with #14.

Or #12 run becuase it was a long run.

You need to be able to show that the WHOLE CIRCUIT is rated for 20 amps before changing the breaker.

The colored cable is a relatively new feature (last 3-4 years), so chances are all the 14, 12, and 10 will be white.

From my limited knowledge of wiring ...

Wider/thicker wire (smaller gauge number) is used with higher amperage circuits because it helps carry the heat from the movement of electricity.

If you run too many amps through a wire it will heat up - just like the filament in a lightbulb - and either melt or start a fire.

Consider adding a second circuit, or plugging some of your appliances into receptacles on a different circuit.