Now that the OP has suffered this twice perhaps some measuring is due this time around. I'd first place a straight edge on the old rim's brake track running across it (or roughly parallel to the spokes). You will likely see that the rim has a concave surface, the straight edge will contact the top and the bottom of the track but the center will be worn in some. How much wear there is is the key. One could actually measure this with a depth gage or feeler gage or thin wire but more to just visualize how much wear there is.

As the rim accumulates miles periodic checking of this wear is easily done. When the wear gets close to the same then...

Another aspect is that as the rim wears the outer edge of the brake track will bulge out with tire pressure. With a higher pressure of a narrow roar tire this will be more apparent then with a low pressure fatter tire. So one could measure the rim's outside width with no pressure when new then repeat over the miles and when the rim begins to get wider with pressure...

Different rim brands and models will have different wall thicknesses so this measuring is only a guide. Andy.