If the mass of a pendulum is increased (with other factors controller), would the period (time) decrease or stay the same?

The period of the pendulum depends only on the length from the pivot to the "center of mass". So if the string has no mass, then the amount of mass on the end doesn't make any difference.

But if the pendulum is suspended on, say, a chain with mass, then the more mass on the bottom, the lower the center of mass is, and the longer the period is.