I am working with a low voltage lighting system on a model train layout. Output of power source is 14volts (amps unknown). 22gauge copper/silver wire to carry load. Wiring completed in parallel with copper side to carry all positive leads and silver to carry all ground leads. Lighting is a mix of DC12v lamps with operating current of 68ma and DC12v LED lamps with resistors attached on positive leads with 3v lamps and an operating current of 30ma. When I connect and activate power from transformer to lights they do not burn at all. I purchased a potentiometer rated at 10ohms with a 20% variable and when 2 posts are connect just after the transformer I have operating lighting for about 60 seconds and then it goes dim and shuts down, I think as a result of the internal breaker in the transformer opening.
There are a total of 14 lamps that are LED with resistors and 16 lamps that are non-LED 12v in the complete line up. My question is what can I do or install to make the lamps stop failing after 45 or more seconds of time? It was suggested to increase the amount of resistance but I don’t know with what or by how much?

What value of resistors are you using now with the LEDs?

Based on the data provided if each incandescent bulb is drawing 68 mA and each LED assembly 30 mA, and everything is wired in parallel, that's a total of almost 1.5 amps. How do you know that each bulb is 68 mA - have you measured their resistance?

One option to consider is putting several LEDs in series with a 500 ohm resistor to create an LED string, and connect each string to the power supply. This will reduce the cuirrent draw considerably, and should still provide enough current through each LED to get them to illuminate.

Sounds like the Transformer may be overloaded/too small, You could monitor the voltage from the transformer, then add loads 1 by 1and if the voltage continues to drop excessively, Then youwill need a larger transformer.
You also may be able to series a few LEDS instead of a dropping resistor for each?
12 Volts is a good voltage to work with.
The dropping resistor is determined By the voltage source and the LED current.
If that set of lights work corectly with that voltage, resistor may be OK.
On a lot of transformers are Listed in VA, Volt/Amps, same as watts, So a 48 VA 12 volt transformer would provide 4 amps. 48/12=4.
Good Luck.

What value of resistors are you using now with the LEDs?

Based on the data provided if each incandescent bulb is drawing 68 mA and each LED assembly 30 mA, and everything is wired in parallel, that's a total of almost 1.5 amps. How do you know that each bulb is 68 mA - have you measured their resistance?

One option to consider is putting several LEDs in series with a 500 ohm resistor to create an LED string, and connect each string to the power supply. This will reduce the cuirrent draw considerably, and should still provide enough current through each LED to get them to illuminate.

Unfortunately I do not know the value of the resistors other than the 10ohm variable I wired into the trunk line just after the transformer.

Sounds like the Transformer may be overloaded/too small, You could monitor the voltage from the transformer, then add loads 1 by 1and if the voltage continues to drop excessively, Then youwill need a larger transformer.
You also may be able to series a few LEDS instead of a dropping resistor for each?
12 Volts is a good voltage to work with.
The dropping resistor is determined By the voltage source and the LED current.
If that set of lights work corectly with that voltage, resistor may be OK.
On a lot of transformers are Listed in VA, Volt/Amps, same as watts, So a 48 VA 12 volt transformer would provide 4 amps. 48/12=4.
Good Luck.

Can I mix series led's and parallel the others on the same line from the transformer?

You can mix series and parallel on the same power source. Let me get you the formula.
Look here:
http://www.hebeiltd.com.cn/?p=zz.led.resistor.calculator
May be some useful info here as well:
http://www.electronicspoint.com/electronics-tutorials-f101.html