We had a storm drop around 5" a few weeks ago and USPS didn't deliver a single letter to the entire neighborhood. Both UPS and FedEx made it through with no problem. Go figure.
In the event of severe weather, as with any company, manpower decreases. The USPS, as well as FedEx and UPS are forced to prioritize deliveries by shipping class.
While USPS may have curtailed their delivery of junk ads, bills, and circulars no one wants, they naturally would have delivered all of the incoming Express and Priority Mail destined for that Zip Code.
Additionally one might want to consider the relative traction of the typical Grumman postal-vehicle (LLV) to a hefty 2-Ton truck which the Private Shippers use as a standard. If I had to choose one, it's the 2-Ton.
Sadly, USPS' supply of 2-Tons is abysmal, thus they are used for things like delivering Priority Mail/Express in massive snowfalls, and daily collection from large mailers and collection boxes.