It is rather more difficult to imagine a situation where an N1 measuring system would give such an over-indication, but even the most inconceivable situations tend to occur eventually!
If however the problem is one of a sudden loss of fan/compressor efficiency, due to FOD for example, then an EPR system will be far more effective than an N1 system as an indication of thrust.
I think think the latter statement cancels the word
"inconceivable'
Takeoff/early climb thrust becomes most critical in a multiple engine bird ingestion. In that case the N1 values will not equate directly to book thrust values by a significant amount.
This disadvantage is not unique to N1 engines since as stated earlier EPR managed engines can revert to N1 when things get dicey.
In either case the pilot should still have the capability to advanced the throttles to maintain his desired pitch setting before the EGT burns out the turbines because there is no real EGT margin left after the engine is allowed to deteriorate for several months/years on-wing..