Is concentration polarisation an observable phenomenon over time in a galvanic cell?
There are two different factors that cause a battery to have a lower energy output. 

First regardless of the "state" of the battery, there is the open cell voltage and the obtained voltage under load. State meaning here how much of the battery is used up. For the open cell voltage the voltage is measured with a high impedance voltmeter so that essentially no current (meaning a very very small current) is being drawn. Next the battery is put under some load typical for its use and the voltage is measured again while under the load. This second load voltage measurement will always be lower that the first measurement for the open cell voltage. This voltage drop is caused by the diffusion limits of the chemicals in the battery to and from the electrodes. 

This behavior can be essentially modeled by assuming that the battery has an internal resistance. So as the current draw increases the voltage drop internal to the battery increases and the output voltage drops. 

Now the second problem is related to the Nernst equation. Let's assume that the battery is used up in 10% increments in a series of tests. At the start of each test the voltage and current is measured. Of course during using each 10% of the battery the reactants in the vicinity of the electrodes are used up and both the voltage and current out for a given load drops. 

Now if the battery is left to equilateral for some time then the reactants redistribute and the battery regains some power. However the battery will not recovery to the level at the start of the previous load test. Essentially the bulk concentration of the reactants is dropping by about 10% after each test. So even after a recovery period both the voltage and current that the battery can supply for a constant load drops. 

Now with all of this being said there are many types of batteries. Some types produce a more constant power output than others.