Again it may come down to the size of the bubble mailer. Some are rather small and maybe thinner and if you are fortunate enough to have your parcel handled by a handler that is in a good mood he might compress it a little or push it through the slot. If on the other hand the envelope is a little larger or has a little more air inside it or your handler is not in so good a mood he/she may just set the envelope over the slot and say, "Doesn't Fit .45cents" Never actually seeing the slot that is mentioned I wouldn't know though, for sure, how thick it is.

On a similar note, I have seen this at the airport with the template on the xray. If you have a soft bag and it won't go through the template because it is about an inch too big and gets caught on an edge, you could be forced to check it. However if you get a screener who is half reasonable that day, they might, or allow you to, compress the bag a little and push it through. Others, might just say "DOESN'T FIT". I had this happen with a hard shell Pelican case once. Went to the UNITED ticket office up the street and got the required dimensions from the agent. Went to the camera dealer and checked the dimensions of the case against UNITED'S specs. Everythig was fine. I bought the case, got to the airport, Just too big. I could have said, Got some sandpaper, and shaved a mm off the feet on the bottom of the case. and it would have fit. The security person said, "DOESN'T FIT, CHECK IT".