This is a real life altering dilemma which has been discussed at some lengths. It seems that not all toilet paper rolls are equal. Some have the perforation cut with a knife-like serrated blade which ensures a perfectly performing tear-off operation.
Others rolls [usually cheaper ones] have their perforations made with a row of spikes, this speeds up the manufacturing. However, this is where the trouble is seated.
Each spike punches a hole in the paper but does not actually cut it, so 'pushing' the paper together between the holes - actually making the so called perforation stronger, like a row of little ropes.
I hope you can now sleep at night. Cheers.