Use the right connector

Ideal Industries makes a few different varieties of twist-on wire connectors that can handle 6 conductors. After looking through the UL Listed Combinations (PDF) document, I was able to find 4 such connectors.

• Wing-Nut 452 Wire Connector, Red. Which accepts 1 to 6 #14 conductors.
• Twister® PRO Wire Connector. Can also handle 1 to 6 #14 conductors.
• Wing-Nut 454 Wire Connector, Blue. Can take 5 or 6 #14 conductors.
• Twister 342 Wire Connector, Gray. Allows 3 to 6 #14 conductors.

If you find yourself needing to connect more than 6 conductors, it's probably time to start looking for something other than a twist-on wire connector. As @Aaron pointed out, you can use a Push-In type connector for up to 8 conductors. You might also want to consider using a Crimp Connector, which can connect 4 to 10 #14 solid conductors (an insulated cover would be required when joining current carrying conductors).

Moving beyond crimp connections, you'll likely have to start looking at terminal blocks, bus bars, design changes, or alternative solutions.

_{^{All conductor combinations will vary from manufacturer to manufacturer, check the manufacturers documentation before using any electrical connector}}

Use devices as connectors

Another option, is to use unused terminals on devices as splice points. For example, you might have a switch with both side and back terminals. Using the back terminal as a splice point, is an acceptable way to extend the circuit. In fact, some industrial grade receptacles (like the Leviton 5252 Series) offer 8 clamp style terminals on the back of the device.

Reduce the number of pigtails

Where multiple devices share a single hot wire, you can reduce the number of wires in a twist-on type connector by using a single extra long pigtail. You'll use the extra long pigtail to connect all the devices, eliminating the need for a single pigtail per device. So you can take the number of required pigtails from 3, down to 1. If you leave the feed hot extra long when originally wiring the circuit, you may be able to eliminate pigtails altogether. To do this:

1. Remove a bit of the insulation in the middle of the wire.
2. Wrap the exposed wire around the terminal screw of the first device, and tighten the screw.
3. Remove a bit more insulation further down the wire.
4. Wrap the exposed wire around the terminal screw of the next device, and tighten the screw.
5. Repeat until all devices are connected.

Once you have all the devices connected in this manner, you can use the end of the wire to feed through to other devices. Simply remove a bit of insulation at the end of the wire, and use a twist-on wire connector to connect this wire to the wires feeding the other devices.

Daisy chain

As for connecting bundles of wires with pigtails. There's no problem doing this, as long as you don't exceed individual connector conductor fill, or overcrowd the box.

Generally, it should never hurt anything to use thicker wire, assuming that you can make solid connections. The wire should never be the limiting factor! The system should already be fused so that #18AWG won't melt, so #14 will only help.

Wire nuts and tape is a tried and true method, although some would argue that soldering and heatshrink is better.

You mention "thickly insulated". Is there any rating information (voltage, temperature, etc) stamped on the insulation? You want to make sure the romex is at least as good.

Specifically, though, it sounds like something else is going on. The existing wire certainly shouldn't have burnt through. The system shouldn't have drawn more current than the wire could handle, and if it did, an upstream fuse or circuit breaker should have caught it. I suppose it's possible that it rusted first, became less conductive, and then burned through...

If there is some sort of unprotected over-current problem, then using #14 might cause the failure point to change. The wire became the fuse this time; who knows what wold have melted if the wire hadn't?

Anyway, just some things to think about...