No. Standard GFI do not trip on 'overloads', that's the circuit breaker's job. A GFI trips on a Ground Fault, hence the name.
A Ground Fault means there is current flowing from the live/hot to ground. Normally, all the current flows from live/hot to Neutral. And current that doesn't return must be going somewhere else, typically ground (but it could also be flowing to the other phase). At any rate, a difference of 5mA between live/hot and Neutral will trip the GFI.
So even though you can have ~ 20 Amps flowing in-out of live/hot and neutral, it's pretty easy/cheap to detect an imbalance of 5 mA (1/40,000th of the main current), so these things are practical and affordable.
I'd guess you've got some crumbs or other food stuff that fell between the live/hot and some grounded part of the case. Is the toaster oven a three prong plug? If not, there is some other path to ground happening, probably NOT your wife, from what I understand you will feel 5 mA as a shock, but it trips fast enough to not harm you (no, I'm not going to test that!). Or maybe the GFI is just getting flaky, and false tripping?
Oh, I can't think of any reason to "wait 3 minutes", this isn't a thermal thing. It's an instantaneous measure of imbalance that opens the circuit, nothing 'lingers'. You can verify this yourself, push the TEST button, then immediately push the RESET button - power is ON, no waiting.
-ERD50