Could a flight ever be considered to be operating under Visual Flight Rules and Instrument Flight Rules at the same time? In the US, could a flight ever be considered to be operating under Visual Flight Rules and Instrument Flight Rules at the same time?

If not, where in the FARs is this spelled out?

Consider the specific case of an IFR-rated pilot flying an IFR-equipped aircraft, but with no IFR or VFR flight plan filed and no IFR clearance requested or delivered, staying in Class G airspace and observing the VFR cloud clearance and visibility requirements spelled out in FAR 91.155.  How can we determine whether this flight is operating under VFR or IFR?  Why could it not be considered to be operating under both?

Bonus question-- is there any specific case, falling within the general conditions described above, where it would matter whether the flight were operating under VFR or IFR?  Assuming that the pilot intends to, and is able to, remain in Class G airspace and observe the VFR cloud clearance and visibility requirements spelled out in FAR 91.155, would it be practical for the pilot to decide in advance that he will simply flip a coin to determine the answer if ATC, or anyone else, enquires whether he is operating under VFR or IFR?  With the added proviso that if no one makes such an enquiry, he will leave the coin unflipped until after landing, at which point he will flip the coin to see whether the flight was conducted under VFR or IFR?  Would such a plan present any practical difficulties?

Related, but not the same-- 

Can an IFR clearance be issued and flown through IMC in class G airspace? -- note that several of the answers, as well as many comments, to this question also address a different question (which probably ought to be asked as a separate ASE question), which is "May an IFR-rated pilot flying an IFR-rated aircraft in Class G airspace violate the VFR cloud clearance requirements without an IFR clearance?"
No, you can't be both VFR and IFR at the same time. It would be like saying that a ball is both red and blue at the same time. You are either one or the other, never both.

If the pilot hasn't received an IFR clearance from flight services, then they are operating under visual flight rules, regardless of their intentions, capabilities, or flight plan. The sole exception is in class G airspace, specifically when they're flying a pre-set departure or arrival procedure from an airport. If a VFR pilot enters a cloud (and there's no emergency in progress), then they are breaking the law, and they can't just decide later that "oh, I was actually IFR at that time".

ATC is never going to inquire about which rule set you're operating under unless there's some kind of administrative foul-up. They're the ones that either did or did not issue you the clearance, so they already know which rule set you're operating under.