You have to use the right size wire nuts, and you have to put them on correctly. If you recall correctly, you have the right size, so you may not have put it on firmly enough. Another common problem is not having all the wires at the same length, so that one is not as well connected. Only when connecting stranded and solid should the stranded be just a bit longer/further into the nut than the solid wire(s).

You may find it easier to use a nut with "Wings" that make it easier to twist forcefully.

Do NOT pre-twist the wires. The wires, stripped to proper length (per instructions of the wire connector - look on the box) and aligned with each other are twisted BY the wirenut as it's installed.

Tape does nothing to prevent or solve a poor connection.

You have to use the right size wire nuts, and you have to put them on correctly. If you recall correctly, you have the right size, so you may not have put it on firmly enough. Another common problem is not having all the wires at the same length, so that one is not as well connected. Only when connecting stranded and solid should the stranded be just a bit longer/further into the nut than the solid wire(s).

You may find it easier to use a nut with "Wings" that make it easier to twist forcefully.

Do NOT pre-twist the wires. The wires, stripped to proper length (per instructions of the wire connector - look on the box) and aligned with each other are twisted BY the wirenut as it's installed (some instructions may say "pretwist optional" but in point of fact I find that it works best if the twisting is all done by the wirenut.)

Tape does nothing to prevent or solve a poor connection.