In intro to QM/Chemistry, I have learned that the number of quantum numbers some object has is exactly the same number of dimensions that it vibrates in. With a real object, there are 3 dimensions of space, and one of time, and so there are four quantum numbers (principle quantum number, angular momentum, projection of angular momentum, spin quantum number).
But, if in string theory there are 9, 10, 11 (how ever) many dimensions, does this mean there are also that many quantum numbers that describe any one applicable object?
Nice question! I wonder, too.
How collapsed does a dimension have to be before it is too small to measure?
How collapsed does a dimension have to be before it is too small to measure?
It is nice to have a genuine question to think about on this board. 
Unfortunately it's not quite as simple as that. The reason that there are 4 quantum numbers for the hydrogen atom is that the electron is a fermion, and thus has a spin quantum number. What I'm trying to say is that the number of quantum numbers depends on the particle you're looking at. If the electron were a scalar particle rather than a fermion there would be no spin. In general for scalar particles there are d-1 quantum numbers, or degrees of freedom, where d is the number of dimensions. A string is different to a particle inasmuch as there is a single direction in which there can be no excitation, therefore a string has d - 2 degrees of freedom.
To answer your question though, the reason physicists talk about string theory predicting 10, 11 or 26 dimensions is by looking at the quantum numbers (actually, the degrees of freedom of the quantised string) in the case of the bosonic string it turns out that the theory makes sense only when the number of degrees of freedom is set to a particular value. To elaborate on that, one way to see what the particular value is is to look at the spectrum of the theory. There is a spin 2 state that we'd like to call the graviton but it is only massless if (d - 2)/24 = 0 which implies d is 26.
To summarise, degrees of freedom are more fundamental in theoretical physics than the number of dimensions.
QUOTE
I have learned that the number of quantum numbers some object has is exactly the same number of dimensions that it vibrates in. With a real object, there are 3 dimensions of space, and one of time, and so there are four quantum numbers (principle quantum number, angular momentum, projection of angular momentum, spin quantum number).
Unfortunately it's not quite as simple as that. The reason that there are 4 quantum numbers for the hydrogen atom is that the electron is a fermion, and thus has a spin quantum number. What I'm trying to say is that the number of quantum numbers depends on the particle you're looking at. If the electron were a scalar particle rather than a fermion there would be no spin. In general for scalar particles there are d-1 quantum numbers, or degrees of freedom, where d is the number of dimensions. A string is different to a particle inasmuch as there is a single direction in which there can be no excitation, therefore a string has d - 2 degrees of freedom.
To answer your question though, the reason physicists talk about string theory predicting 10, 11 or 26 dimensions is by looking at the quantum numbers (actually, the degrees of freedom of the quantised string) in the case of the bosonic string it turns out that the theory makes sense only when the number of degrees of freedom is set to a particular value. To elaborate on that, one way to see what the particular value is is to look at the spectrum of the theory. There is a spin 2 state that we'd like to call the graviton but it is only massless if (d - 2)/24 = 0 which implies d is 26.
To summarise, degrees of freedom are more fundamental in theoretical physics than the number of dimensions.
In the above, I stated "(d - 2)/24 = 0 which implies d is 26." This should read (d - 2)/24 = 1 of course.