Based on the technology in my answer in Hard Sci-fi energy shields:
The technology is based on a swarm of physical shield elements held in place and moved around using superconductive flux pinning.
There are two resources that can be depleted: the physical plates and the control system.
Plates that hit may break, or undergo stress that cause fractues to build up with use so it eventually breaks. They are designed to absorb more energy by ablating a passive coating so this wears off. It also absorbs energy by breaking along engineered seams so the smaller pieces can still be contolled by the system.
Plates may suffer a hit and absorb more energy than the flux pinning system can “catch” so it’s thrown out of the swarm.
In any case, fresh plates need to be deployed to continue to provide protection. It takes time to move them around and you may have a control capacity of only moving so many at one time. Bad ones still consume control capacity so must be removed before fresh ones can be put out.
When a plate is pinned in position, any force acting on it induces electric currents that set up magnetic fields that exactly counter the motion. When hit the plate is accelerated severely and thus causes huge currents. The superconductor has physical limits.
To reposition a plate requires active control over the currents, essentially making an electric hologram that changes over time. How many plates can be controlled at once, how quickly? There will be reql limits to the machine.
Imperfections will consume more power and cause heating.
The boss wants a single number that indicates how much impact can be accepted, from a specified direction. You can see that after being used, plates need to be sent back to the correct positions or rotated out of service. Heat must be removed from the superconductor, and power added to carefully dampen/cancel disturbances and reposition the plates. Until that is accomplished, the system will have a reduced maximum ability for the next hit.
This is a summary, but is really an assessment of many factors. So it may indeed be inconsistent in how it really behaves. It’s not a simple hit-point scorekeeping, but a complex dynamic state and messy fracturing of physical plates. How it really handles the next hit can be quite variable.
So when the capacity is exceeded, what happens?
If there is no plate that can take the hit, or deflect it completely, then the impact will hit the ship. This may be leftover bits and pieces that could not be slowed in time, so the effect can ramp up as the capacity is exceeded.
More spectacularly, what happens if the superconductor is overloaded? Look what happened at the LHC when it “quenched”. The heat build up caused it to exit the superconductive state, and then the entire energy stored as electric currents suddenly felt the material resistence.
The superconductor shell doing the control needs to surround the entire ship. It might be seggregated into individual units that have some isolation from each other. When one of them blows up, it’s like a grenade exploding right at the skin of the ship! So, this matches the TV trope quite easily.