Why do different resistors of the same power rating burn out / not burn out?
To determine the actual power dissipation you should measure the voltage across the resistor when powered on. 

The power dissipation is then \$V^2/R\$. The most voltage that a 47 ohm resistor can have and still be within the 1/4-W rating is 3.4V. 

It does not matter what the current capability is of the supply (and you should never put an ammeter across a voltage source because you could damage the source or your meter), what matters is what the voltage is under load and the resistance. 

If you have multiple resistors in series or in parallel, each will have it's own dissipation depending on the resistance and the voltage across it. 

When you use (or abuse) the LED the voltage across it will change so the voltage across the resistor(s) changes from open-circuit, and of course you are also fiddling with the resistance.