Depends on the sample height you instrument can read upto. For example, in the Bruker Advance series of the spectrometer the total volume of sample is kept between 500-600 microliter.
10 mg of a pure sample should be able to give you a decent 1H and 13C spectrum on a 400 MHz NMR. if using normal size NMR tubes, try not to exceed 700 microliters with the deuterated solvent as excess may further dilute the sample leading to smaller peaks.
If however you have smaller samples, try running longer 13C scans to obtain decent peaks. Also, you could try using smaller NMR tubes for smaller samples.
Generally 10mg of pure sample in ≤ 700 microlitres of deuterated solvent should give you decent peaks
The smallest amount you can observe in NMR is probably around 0.5-1 mg that is dissolved in 0.5 mL of solvent. If your molecule has higher molecular mass (>500) then your peaks will be even harder to observe.
It is recommended you do 256 scans if you are working with small amounts of material.
We get this question a lot. The answer is: "it depends". It depends on many things. How complex is your molecule? What is the multiplicity of the signals? What is the field of your spectrometer, what sort of probe do you have, what is the solubility of your material, how pure is your material, what sort of data do you need, how long have you got on the instrument, ...? The list is very long. The answers here show some options.
Generally, you will get a decent 1D proton spectrum on a pure compound of around 300 mw at 400MHz with a conventional inverse-geometry 5mm probe in 32 scans on 1mg of material. You can push this to lower amounts through the use of smaller diameter tubes and more scans. Don't forget that the signal to noise increases only with the square root of the number of scans so half the material will take 4 times the number of scans (= time).
If you have it, more material is better and you can always get it back as NMR is non-destructive.
As John mentioned, it depends. Mostly because molar concentrations of the active nuclei matter in spectroscopy, not the w/w or w/v. So, 0.1 mg/ml may be pretty ok for 1H NMR of a small molecule, but not the large one and not (natural abundance) 13C NMR.