The most common thing that can cause proton signal broadening and an increase in peak integral value is exchange broadening between two different protons that are slowly (on the nmr timescale) exchanging in your sample. For example, if you have an alcohol $\ce{-OH}$ signal, but there is also some water in your sample, then the protons will exchange between these two environments. The observed signal will be broadened and have a larger integral value than expected. Also, the position of the broadened peak will be a weighted average of the chemical shifts for the two independent signals.

If adventitious water is the culprit, you can add a small amount of $\ce{H2O}$ to your sample and see if the peak shifts and the integral value increases.

The most common process that can cause both peak broadening and an increase in the value of a peak integral is exchange broadening between two different protons. The following diagram represents a typical case where the protons in water are exchanging with the hydroxyl proton in an alcohol.

If the rate of exchange is fast on the nmr timescale, then a single, sharp peak of increased amplitude and a weighted average of the individual chemical shifts is observed. If the rate of exchange is slow compared to the nmr timescale, then two distinct peaks, both with accurate integrals, are seen. In the case where the rate of exchange is comparable to the timescale of the nmr experiment, then a broadened signal with integral intensity equal to the two combined signals and located at a weighted average of the two separate signals is observed.

One way you can test to see if a small amount of water in your sample is the culprit is as follows. Add a small amount of $\ce{H2O}$ to your sample and see if the peak for your sample shifts to a new location and the integral value further increases.