Confusion about resting membrane potential and the Na/K pump
The Sodium/Potassium pump maintains the overall concentrations of ions on either side of the membrane. In other words it keeps the neuron's battery charged. 



Imagine the neuron like a circuit. We can imagine that the electro-chemical gradient for each ion species (Potassium, Sodium, and so on) as batteries. So When the ions flow according to their electrochemical gradient, the potassium current flows outward and sodium  current flows inward as dictated by these batteries reversal potentials [negative for potassium, positive for sodium]. 

However, note that this will discharge those batteries, i.e. the concentration of sodium on the outside of the cell will drop (slightly) as sodium flows inward the cell, and potassium flows outward the concentration on the inside will drop slightly. Thus the magnitude of the reversal potential of the Sodium and Potassium drops (they move closer to zero). 

To counteract this discharge the Sodium/Potassium pumps expends energy in the form of ATP in order to recharge the batteries. The amount of current that the pumps moves is very small relative to the larger currents of sodium rushing in and the potassium rushing out during an Action Potential. Thus when a neuron fires there is very little overall current contribution from the pumps. 

When the neuron is at rest (not firing) then the Sodium/Potassium pump is able to recharge the membrane and the current becomes important. Still this is a very slow process, relative to action potentials.  

Another thing to note, even though the cell is at rest the batteries are discharging slightly. Their is a leakiness to the membrane, sodium leaks in and potassium leaks out. Thus the pump's current will counteract these leaky currents at rest, keeping the batteries charged and thus the overall net current  across the membrane at rest will be zero.  

Remember though, these currents are very small currents. 

EDIT: Based on Discussion with @Christiaan, I decided to post a update to clarify.

At rest potential Sodium Ion influx is much smaller than potassium ion efflux and thus can be ignored. Whether or not Sodium is ignored the above still holds true. The neuron's membrane is at rest which means the net current is equal to zero. This means that the outward potassium current is counterbalanced by the pump's current thus keeping the relative concentrations of Potassium and Sodium constant while at rest and maintaining a steady rest potential.  

EDIT 2: Clarifying as per OP comment:

So during an action potential Sodium and Potassium ions move in large currents and this current flow discharges the batteries. At rest there is still current flowing (thus smaller discharging of the battery) but there is no Net current flow thus no change in voltage(thus the neuron is at rest). It is exactly counterbalanced by the sodium/potassium pumps. The reasons the pump doesn't overcharge the batteries is the constant but smaller discharge at rest. When the recharging current is equal and opposite to the discharge via the leak current, a equilibrium is reached.

However when lots of action potentials fire, the resting potential will be altered from the larger discharge rate, thus when the neuron returns to its new resting potential the leak current and pump current will not be balanced, thus the recharging pump current will be larger in magnitude and will bring resting potential back to its normal value.  (Note what constitutes large is relative between species and even neurons.) 

Edit 3: Rates of Leakiness and Pump activity.

If only you could say one is stronger than the other. It depends on the conditions of the cell.

Lets focus in on potassium. The leak current for potassium is dependent on two (really three) things. The permeability (or conductance) to the membrane to potassium, and the reversal potential of potassium (which is dependant on the internal and external Concentration of Potassium. Thus varying these things will strengthen/weaken the power of the leak current. 

The Sodium Potassium Pump is dependent on three things. The External Concentration of Potassium, The internal concentration of Sodium and the availability of energy (in the form of ATP). 

The point where these two strengths are about the same is when there is more Potassium on the inside of the cell. That is to say pump is capable of keeping up with the leakiness of the membrane until the internal concentration of potassium reaches about $140$ $mM$ (again that just a ballpark number). If their is less Potassium in the cell then $140$ $mM$ the pump is stronger, if there is more than $140$ $mM$ the leak of potassium is stronger.