Upgrading my XPS 15 (L502X) RAM

You can run 16 GB RAM in that system. I have the identical laptop using 16 GB.

I hope you bought a matched pair of RAM, as that is more likely to lead to success. DDR3 1333MHz definitely works. DDR3 1600MHz may work.

I also hope that you used a grounding strap when handling the RAM and installing. Without one, you could cause an invisible static discharge that damaged a RAM stick or slot.

You need to narrow down if the issue is with one stick of RAM, one RAM slot, or something broader.

First, rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard. Wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. I know many people do such work without this, but it's a cheap and sensible precaution.

Disconnect the AC adapter.
Disconnect the battery.
Press and hold the power button for 30 seconds to discharge any residual electricity.

Remove the RAM sticks.
Use a soft rubber eraser (the type used to erase pencil) to clean all the RAM contacts on both sides. Use a clean cloth to brush off any eraser residue.
Moisten a microfibre cloth with 99% isopropyl alcohol and use a credit card to slip the damp cloth into each RAM slot to clean its contacts.

After everything is dry, insert only the suspect RAM stick firmly at a 30° to 45° angle, ensuring that it seats fully in the slot.

Reboot and see if it works.
If it doesn't, then:
Disconnect the AC adapter.
Disconnect the battery.
Press and hold the power button for 30 seconds to discharge any residual electricity.
Move the suspect RAM stick to the other slot.
Reboot and see if it works.

If the suspect RAM stick won't work in either slot, and the other RAM stick works in either slot, then one RAM stick is bad.
If either RAM stick works in one slot but neither works in the other slot, then that one slot is bad.

You didn't say whether it has a hard drive or an SSD. If it's a hard drive, replacing it with an SSD will make it much faster and more responsive, a noticeable difference and the most bang-for-your-buck that you can get for any upgrade. I have a Crucial Mx500 1TB SSD for mine, and Crucial recently reduced the prices on that excellent line of SSDs.