Potentiometer value goes up and then down again
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. The equivalent circuit.

You have 15 x 4k7 pots in parallel with the one you are testing.That means that there is a resistance of 4k7 / 15 = 313 &Omega; between the rails due to the other pots. If you measure resistance between the wiper and ground and rotate clockwise your should go from zero at the bottom to 313 &Omega; at the top.

In the centre you will have 2.35 k&Omega; to the bottom due to the bottom half of the pot in parallel with 2350 + 313 &Omega; due to the top half of the pot plus all the other pots. This should result in a reading of about 1248 &Omega;.

If your soldering is OK then turn on the power and you should get a linear voltage increase from 0 to 5 V as you rotate the pots.