I've seen a couple companies (credibly) claim multiple independent networks, but it seems to be pretty expensive to do in practice. There's a lot of cost to be saved by having a single network, and it's too tempting to share things making the independence illusory.

Probably you get better independence with smaller providers where the provider isn't yet big enough to actually run a global network. Then, each PoP is likely to run or fail independently. Google and peers have enough scope that they can run a single global network where one can experience the Haiku:

It’s not BGP There’s no way it’s BGP It was BGP

(DNS also fits in there; any sort of automation to push DNS or BGP changes will probably fit, if the name was carefully chosen)