The harmonic series is not a sequence of powers of two. It is a sequence of integers times the base frequency. Using an open A440 as an example:
1x440 = 440 The fundamental of the series
2x440 = 880 The first overtone of the series and also the first octave
3x440 = 1320 The second overtone of the series, but not an octave
4x440 = 1760 The third overtone of the series and also the second octave
etc
So there are many overtones of the harmonic series that are not octaves of the fundamental frequency.
As Don spectral plot proves experimentally, the violin overtones, for all practical purposes, are integer multiples of the base frequency. So experiment and theory match.
I would take technical exception with Michael Darnton's observations about "computed" FFT frequencies. FFT does not actually compute frequencies. Rather, it computes the contribution of each frequency across a large and detailed frequency spectrum that depends on the sampling rate (how many times per second the analog tone is captured as number) and the time span of the entire sample.
If you convert a violin tone into a digital signal using a high sampling rate for several seconds, then the FFT algorithm can accurately compute the contribution to the signal of all frequencies in the spectrum range down to a very fine frequency resolution.
Inharmonicity is the deviation of overtone frequencies from integer multiples of the fundamental. Effectively, this does not happen for a "well bowed" violin tone. If it does occur, a well sampled tone would display peaks at frequencies that are not integer multiples of the frequency of the first peak.
It is possible that tones can sound "false" if the violin does a poor job of reproducing the fundamental and some of the lower octave overtones, or does a much better job reproducing the higher overtones than the lower ones. The overtone frequencies are accurate, they are just not very strong.
I think what you are looking for is some general observation on what the strength of each overtone should be relative to the fundamental. Call this the overtone profile.
I studied this a few years ago by sampling tones from various recordings of highly regarded violins. The note needs to be sustained for a second or so and played with no vibrato. Where the violin bow point was relative to the bridge affected the profile. The profiles of a good G string were quite different than that of a good E string.
When comparing similar plots made of bad violins, I could usually tell why a bad violin sounded bad, but could not deduce any reliable rule as the why a good violin sounded good.
Even if one could come up with a set of overtone profiles that many could agree gave a beautiful violin tone, it is not obvious to me how one can translate that into practical rules for making a violin.