60watt max means it will burn something up if you go higher. If it takes two bulbs, and you're happy with one brighter bulb - perhaps use 2 40 watt bulbs. It will be safer.

Frequently wattage limitations are imposed for enclosed fixtures where the bulbs are not subject to open air circulation. If the fixture has two sockets and you only use one bulb the light may be "one sided".

To state it another way, about 96% of the wattage of an incandescent lamp is converted to heat. The limitation on wattage in a fixture is not related directly to a problem with higher current draw, but with the heat resulting from the lamps used. I am of the opinion that any deviation from that specifications will result in shorter lamp life and shorter fixture life. Insulation on the conductors in the box serving the fixture might be damaged as well.

To follow on Bus's comment, heat is proportional to wattage. I would expect that any combination of lamps summing to the maximum specified (in this case 120 watts) would be safe.

An interesting side comment : a single 120 watt lamp (if such a creature existed) would produce more light than two 60 watt lamps. All other things being equal, the higher wattage lamp is more efficient at producing light. Check the lumens/Watt ratings of various wattages of 120 volt, standard service bulbs, to see what I mean.

If you want to be really safe (and save some money on electricity in the bargain), fit a screw-in compact fluorescent. It will produce much less heat for a given amount of light.

There are two limits here.
The sockets themselves may not be able to take a higher wattage individually, and the fixture may not be able to take a higher combined wattage.
Without knowing what component limits the wattage you should use individual bulbs no larger then the marking

you could use the twisty fluorescent light bulbs. Im thinking the 100 watt version only uses like 20 watts or something