Highly unlikely that the knotted cable would create enough inductance to have any effect whatsoever on a voice phone line. Even to affect normal voice frequencies (which are higher than the ringer current) significantly, you'd need a coil with hundreds of turns of wire.
If the cable itself is defective, and has broken conductors inside, knotting it might possibly pull the broken wires apart enough to open the circuit. Or, you might have a defective plug or socket, which sometimes doesn't make good electrical contact.
Does one, or more of your old phones have an honest-to-goodness bell in it - a real mechanical bell with a "clapper"?
Those old-style ringers can take quite a bit of current to operate correctly - much more than the simple piezo-electric "chirpers" used in more recent phones. The bells and their solenoids were designed back in the day when the central phone office was responsible for sending the "ringing current" down the wire - this was a relatively high voltage and the ring generators were capable of delivering quite a bit of current.
It's not unusual to find that new-style ATAs ("analog telephone adapters") have much weaker ring generators in them, and can't source enough current to ring the old-style bells reliably. This can affect both single-purpose ATAs, and the ATAs build into cable modems and similar Internet interfaces.
Here in the US, each phone normally has a label on the bottom giving the REN (Ringer Equivalence Number) for its ringer. A value of 1.0 means "as much current as a standard Bell telephone bell". If I recall properly, phone companies here are supposed to be able to deliver up to 5.0 RENs for a normal phone land-line. "Chirp" ringers are typically in the 0.1 - 0.5 REN range, I think.
An ATA may not even be able to deliver 1.0 RENs. It depends on the brand and model. The owner's manual may tell you what the limit is.
If you have two or more phones hooked up at the same time, you have to add the RENs together to figure out the total ringer load.
I've seen cases in which bell-type ringers would work sometimes, and not other times, even though nothing had apparently changed on the line. If I recall correctly, there turned out to be some residual magnetism in the ringer solenoid, which sometimes caused the ringer to physically "stick" and not vibrate freely.
Check to see if your cable modem's VoIP gateway software has a call log available. This would let you check if the gateway even saw an incoming call at all, at the time that your mother tried to call you and the phone didn't ring.