I don't know why it would only happen when another burner was turned on unless it was routed in close proximity to it. Power should not be a factor in that symptom only heat.
Typically wires on appliances burn for 3 reasons:
1. A poor connection between the wire and the wire terminal.
2. A poor connection between the wire terminal and whatever it attaches to (switch, element, etc.)
3. A poor contact inside whatever the wire is attached to.
These poor contacts generate heat with is transferred to whatever they are physically attached to affecting them as well.
If this happens on a plug-in surface element right at the receptacle that the element plugs into, a common cause for that is described at the following link: