Any mains rated connector has to pass certain insulation tests to comply with the standards. No doubt it will work at much higher voltages, than 250V with no problems, but it isn't certified to be safe, at such voltages.
The voltage rating also assumes a level of contamination commensurate with the connector's intended environment.
I was looking at this connector:
https://www.te.com/commerce/DocumentDelivery/DDEController?Action=showdoc&DocId=Specification+Or+Standard%7F108-40018%7FF%7Fpdf%7FEnglish%7FENG_SS_108-40018_F.pdf%7F5499923-9
It says it has a voltage rating of 250VAC (which I assume is 0 to pk). However, it also has a series of tests done that are in excess of that voltage. The insulation resistance is a certain value....but I am guessing they are using air as the insulator? The next value is withstanding voltage, but from what I read, that is more to do with high voltages due to lightening strikes or inrush currents.
So, my question is what happens when I exceed the 250VAC rating? I am guessing that it begins to conduct to a neighboring pin? Or is this more a function of creepage and clearance, so the distance between pins determines the rating?
The 250v AC rating will be RMS, which implies a peak voltage of around 350v.
In the real world, voltage ratings are conservative, so a small % increase in voltage will do exactly nothing.
I wouldn't try to use them at 400v RMS, though!
DC is something else again.
With its propensity for arcing, AC rated connectors may not be useable at their marked voltage or specified current ratings.
"Or is this more a function of creepage and clearance, so the distance between pins determines the rating?" - yes, it has to withstand overvoltages like portion of lightening strike and so on and not to make permanent arc surfice short (arc have to die).
I mean the problem is not working voltage but overvoltages and connector is rated for them.
That is a large difference with semiconductors which are rated to withstand only 20% higher the rated voltage.
Voltage ratings for wire and connectors tend to be very conservative at room temperatures. As an example, most RG-8 (with the solid PE dielectric, not the foam) will hold off 60 kV without issue. At 105 degrees celsius it may be a different story.
The point of the higher voltage tests is to make sure that the connector won't arc over due to a high-voltage transient. This can be caused by lots of things besides lightning such as the back EMF from a large inductive load.
You'll also need to considerably derate for DC use. DC arcs don't self extinguish like AC.
Not sure if this holds for all items within a circuit, but this is generally how we do it for connectors...
Step 1 - establish the maximum voltage (withstand voltage, flashover voltage). You don't test it all the way to full flashover, you just establish the point at which you see a leakage voltage to the next pin over. Testing is done with a mated connector pair, to see when the leakage occurs to the next mated pair (or metal hardware if that is closer, although unlikely). Maximum voltage will be set just under the leakage voltage, normally to a nice round number.
Step 2 - Voltage rating is calculated from this maximum voltage. This is basically set at a level that is a percentage of the maximum voltage. Some of our connectors are set at 1/3rd of the maximum, a couple are at 2/3rds, but that's the highest.
Now, our connectors are not aimed at the household appliance market - I would think these connectors would be rated at 250V even if the max came out at 1000V - although the max is far more, 250V is relevant to this market. However, they might also rate it at 250V even if the max is 350V - I would expect that there is probably some legislation or standard that defines a required minimum amount for the max voltage. But that is totally an assumption on my part, so please don't assume this is valid for your item. Which means none of this is particularly helpful necessarily, just thought it might be useful information!