More important is that you have large enough wire to carry the current. If you upgrade a breaker but have too small of wire going from the receptacle to the breaker, the breaker won't trip but the wire can/will overheat with potential fire issues.
Sorry if I'm getting off the beginning welder topic but this seems like a relevant point. My question regarding wiring is this: my service panel is pretty far from where I need to put my welder - like 60 ft. Do I need to run even heavier gauge wire due to the distance from my panel to my welder? I assume there's the similar drop in power you see with running long extension cords.
I don't see how tripping a breaker once in a while would hurt the welder.
And don't worry too much about other loads in your panel except those that might run at the same time as you would be welding - lights, maybe a heater ....