Note: See my companion articles on Diagnosing Car Electrical Problems and Understanding the Check Engine Light.
In many older cars, people run into mysterious electrical gremlins that cause no end of grief. However, electrical circuitry is not magic, but science. Electromagnetism is magic.
The general goal of any electrical system is to minimize the number of connectors and breaks in the system, as each is a source of potential trouble. In this regard, cutting and splicing wires should be avoided. Some folks, to try to save a few dollars, will try to cut and splice a "generic" oxygen sensor into the existing harness. This may result in a savings of maybe $60 per sensor, but involves quite a bit of work, to do the solder job right. And each connection is a potential problem (cold solder joint, water ingress, corrosion) that could make the job go horribly wrong.