|MadSci Network: Physics|
For multi-dimensional string theories with 3 extended space dimensions, 1 time dimension, and a bunch of other compacted space dimensions... it looks like your friend is right. Less than 10 dimensions gives nonsensical mathematical results, and more than 11 predicts that new massless particles (with spin >2) should exist. Becase these particles are not seen in nature, one can conclude that 11 dimensions is the upper limit in order to match up the theory with our universe.
But, as always, there are caveats.
First off, once can consider adding not spatial dimensions, but another time dimension! This seems to be allowable... but it's not a very popular theory. I don't know if there's an upper limit on the number of time dimensions or not.
Secondly, I'm not sure if the mathematics applies to universes where more than 3 spatial dimensions are extended. You might have noticed that we don't live in a universe with more than 3 extended dimensions, but some very recent work has hinted that maybe other infinite dimensions are out there, but we just can't see them! If this is the case, the above dimensional-limit might no longer apply.
And finally, there's the caveat that nothing in physics is ever certain, and the mathematical basis for the above conclusions could be supplanted some day by a different theory which DOES allow for more dimensions. String theory is a quickly-evolving field (the jump from 10 to 11 dimensions was only made in 1995!), so many questions are still open.
If you're interested in this topic, an excellent non-technical book is Brian Greene's "The elegant universe".
Try the links in the MadSci Library for more information on Physics.