Corey-Schmidt oxidation of primary alcohols
As stated in the previously posted answer:


  This oxidation into
  carboxylic acids succeeds when the intermediate aldehyde equilibrates with a liberal proportion of hydrate; that is, when the intermediate aldehyde belongs to the aliphatic kind and is not highly hindered. The water necessary for the formation of the intermediate aldehyde hydrate may proceed from the decomposition of PDC.$^{[1]}$


Though the text does state that water may form from decomposition of the dichromate anion, since DMF is so hygroscopic as a result of its high polarity ($\epsilon_\mathrm{r}=38^{[2]}$), I would not be surprised to learn that little to no over-oxidation occurs in completely dry DMF under an inert atmosphere.

This is only speculation though, and the dichromate anion may very well be decomposing to produce water in the solution. As such, I will propose a reaction mechanism for this decomposition. The first step can only occur when pyridinium dichromate is well solvated and protons are free to be transferred between the species, which occurs in DMF but not in DCM:

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This produces chromic acid, another chromium(VI) oxidizing agent. Unlike dichromate oxidation, however, chromic acid oxidation actually produces water as a byproduct, which is then free to hydrate the resulting aldehydes:

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$[1]$ Tojo, G.; Fernández Marcos. Oxidation of Alcohols to Aldehydes and Ketones: A Guide to Current Common Practice; Springer: New York, NY, 2006; p 28.
$[2]$ Wikipedia: Solvent