[old_tv_nut]

The resistor in parallel (if the peaking coil has one) affects the bandwidth of the peak (or null, if it is used in a trap circuit). The DC resistance of the coil is usually much lower than the parallel resistor, so the resistor value could change a lot and you might not notice the effect. Coil failure hypothetically can be either opening, or shorting of one turn to another.

I think that these low power peaking coils usually fail open due to corrosion over a long period. Coils and transformers that carry high power (like in the deflection circuits) can fail due to heat or arcs causing shorted windings. In this case, the AC operation is ruined, but the DC resistance may not change much, because only one turn is shorted out of tens or hundreds.

Maybe some others can chime in on which types of failures typically occur in different coil applications.