The ones I've pulled used a utility knife blade to cut the string. If the blade falls off or if the tying arm doesn't travel quite far enough it wont cut. The tying arm could be slightly bent or mis aligned.
Depending on the type of baler you have there is a adjustable twine stop located on the right hand side of the baler located right in front of the twine arm. We've come across this type of problem on our 605F. When the twine arm is up in the cutting off postion the twine should be snug against the stopper. We use plastic twine. If you use sisal the stop adjustment may need to be fine tuned since it is of a different size. We've put thousands of bales through our machine and have never replaced a blade or sharpend. Also check the twine tube for wear especially by the twine stop.