Dissolved oxygen is the most common paramagnetic impurity that most people are likely to encounter. It is best to flush out the dissolved oxygen with nitrogen gas (preferably helium) - incomplete removal of oxygen can certainly degrade the quality of the spectra.
Another source of degradation is the presence of particulate material consisting of suspended aggregates - dynamic motion of the sample between the aggregate phase and soluble phase can lead to considerable line broadening. This effect can be reduced by passing the sample through a filter. A sample containing such aggregates may look pure in LC-MS - however, the sample preparation part of NMR involving concentration or solvent exchange or lyophilization may cause precipitation/aggregation. Some samples may aggregate over time, i.e., they are fine when passed through the filter, but the aggregates appear at a later stage. Dynamic light scattering should be able to measure the extent and size of these aggregate particles.
Here is an extremely unusual case:case (dealing with a protein; not with a small organic molecule):
Once I was working on a 15N-isotope labeled protein that seemed stable enough outside the NMR spectrometer, in the NMR tube; however, whenever I tried to obtain a high resolution HSQC spectrum at the same temperature (even with the broadband decoupler off, to prevent heating) it would degrade!! (I know that it was not degraded at the start because a quick low resolution HSQC at the beginning clearly indicated a clean folded protein). It was a small protein, so it is unlikely that the magnetic field caused any partial ordering/alignment. Also I incubated the sample for different intervals outside the spectrometer in the NMR tube - there seemed to be no evidence of degradation - it appeared degradation started only when the spectral acquisition process was initiated. (The work was never published, and we have not checked that these results are reproducible, because of the obvious effort and cost involved in preparing the 15N labeled protein). Although I could not propose any specific physical mechanism it appeared that it was RF induced degradation!
Note: I have seen reports of the effect of RF irradiation on cells and other living systems - but none of these reports indicate the possibility protein degradation so extensive such as the one that I have observed.