If a mould blown piece is put onto a punty for finishing, changing shape, adding bits, it will be removed from the blowpipe. While the finishing is done the edge will also be fished using a tool and probably a bit of fire polishing. After cooling the scar left by the punty is then ground down and polished (as appropriate) - to then go to the additional cost of cutting the rim would rarely happen and probably only when a metal rim is to be added.
However, after removing from the mould on the blow-pipe various additions can be made without transferring to a punty. As the base is mould finished there is no need to grind or polish it. But when the piece is removed from the blowing iron the edge will be rough and sharp. This gets ground and polished.
Blown cased glass that has a fire-finished rim is the norm for punty finished and the layers of glass on that rim tend to blend. As soon as it has been ground to remove a chip, the layers are very visible (If you know what to look for). Some say a good restorer can grind and polish a rim to appear fire-ground - in my experience this is not true unless you do not understand what to look at.
Sometimes you will find blown glass with both punty or gadget marks on the base and a rim that has not been fire-polished, just sheared while on the punty and tidied up by grinding. The only examples of this, I have seen, have been a few pieces of free-blown non-standard production Strathearn.
So as Ivo says it is a warning flag and in the world of variety of glass it is not impossible for some examples to exist. But they will be rare and unlikely from a 'cheap' production where costs are minimised.
Another case might be where it has been mould blown and a separate foot applied.
And of course, cut glass.