Question: Because, physically, if the volume is constant and the pressure increases, won't it mean that there is an increase in that internal energy?

If not, how else should we perceive it?

Answer: 
If the volume does not change, then there is no work being done on or by the system. Any change in internal energy would be associated with the process which caused the change in pressure: heat or gas being added or removed.
For a single phase fluid of constant composition, the change in molar internal energy can be determined for any arbitrary variation of T vs P by integrating the equation $$dU=C\_vdT-\left[P-T\left(\frac{\partial P}{\partial T}\right)\_V\right]\left(\frac{\partial V}{\partial P}\right)\_TdP$$This integral will depend only on the two end states, and not on the chosen variation in T vs P used to evaluate the change. You can see that the second term is definitely not VdP. The integrand in the second term depends on the equation of state for the fluid. For an ideal gas, the integrand is zero for all variations of T vs P.