You can run electric motors far above their continuous rated values for short periods. Also, electric motors specifically designed for brief overloads (high-temperature insulation, temperature sensors) are quite possible. Every automobile starter motor is such a motor. With synchronous brushless motors ("brushless DC" and "variable frequency synchronous" motors are the same thing; motors above a few KW tend to be called the latter) and big power IGBTs, you can have huge torques briefly without much difficulty. If you have the electric power available.
Batteries, maybe. Running time between charges is going to be a problem for a long time to come. There's a huge battery industry trying to get energy density up, with modest success. For many applications, trailing a power cord most of the time is an option. Especially if the robot can plug itself in, which the Hopkins Beast was doing in the 1960s.
[1] http://www.gizmag.com/siemens-world-record-electric-motor-ai...
The problem is not power density, but torque density. Brushless motors spin really, really fast with low torque, which is the exact opposite of what we need for robots.
SCHAFT found a solution to the more current problem with their ultracapacitor driven water cooled motors. Except one cannot drive said motors continuously and alone they still don't have that much torque, so if one wants more torque more windings must be added.
You can design motors for higher torques at lower speeds but the torque density suffers. Luckily we have compact high reduction gearing to transform high speed low torque to low speed high torque.
If the charging time was close to zero, would this problem be solved, for many applications ?
Or the other alternative, phinergy's aluminum air battery, which has 2000 wh/kg, but cannot be recharged, just replaced and "recycled", but probably in a cost effective way ?