Why would cat6 connectors not work with cat5e patch cable

You don't say how you've bundled them together. Did you use zip ties and pull them tight? Did you loop the cable and cinch the loop (like a figure eight)? Are there any other places where there are sharp bends? How small is the radius of the loops or bends?

You don't say whether they're pulled through walls or drop ceilings. You don't say what the conditions are that you're working in. Stray nail tips and sheet metal edges, etc. Are you running parallel to any fluorescent lights?

Besides squirrels and rats (which can be very real problems), most of the time it's rough handling or improper termination that cause problems.

You say "I punched a few jacks on the end of a length of cable". I presume that's just an imprecise way of saying "I punched one jack at each end of a length of cable" because you can't punch a few jacks on a cable for networking and get away with it like you might for multiple phone jacks on one cable.

Also, continuity and "polarity" mean essentially nothing. Your test with two computers or a computer and a switch or best-bet rent a proper Cat 5 or greater tester will be the only ways to approach testing. Lots of things conduct current that won't work for networks.

Also, don't trust patch cables. Try another one.

It isn't the cable that's the problem. That's just a matter of insulation and manufacturing tolerances, the actual cable pairs are the same with either cable. The important bit is what protocol you're going to running over the wires. Some protocols only use two pairs, others use all four, and obviously you'll only be able to use a splitter if you're running a protocol that only requires two pairs. Also, not all the pairs are created equal (from a twists-per-metre perspective), and so again, depending on protocol and NIC, you may or may not have problems with cross-talk.