Check those brake pads. On a conventional caliper (dual pivot, side-pull or center-pull) the pads move up as yhou squeeze harder. Make sure you can still see rim above the pad everywhere on the rim when you squeezew hard. (You may have a low point on the rim where the pad slips over, rubs and weakens the casing.

Also run your finger along the rim at the blowouts. All smooth? If not, time for a little fine sandpaper or file.

Ben