What you will want to avoid is the planet going into negative population growth because there isnt any room for anybody to bend over to pick up the newborns that are then trampled.
I am reminded of Larry Niven's books, "The Mote In God's Eye" and "The Gripping Hand."
IF a society can NOT damp down its citizens' drive to procreate in ever increasing numbers, the day will abruptly come when the stocks of necessary resources are insufficient to meet the demand adequately. When that translates into large scale die-off from starvation and inadequate health care, the Have-Nots WILL rise up and take from the Haves. The resulting population reduction (French Revolution, Russian Oktober Revolution) will cause something of a calming effect as the survivors work out a stop-gap political solution that may or may not reconfigure the system to function more effectively.
HOWEVER, we are dealing with interstellar societies. Whatever interstellar government is in control will NOT condone something that disruptive. (Either because of the general citizenry's willingness to help out on the premise that, "We're all in this together," or because of the government's concern that such disorder may spread to other planets in the empire. The upshot of that is that _something_ would be worked out to compensate for that planet having hit its population cap. And that something has very limited possibilities:
1) Shifting population to less crowded areas (other planets).
2) Draconian birth control (serious drop in Morale).
3) Tech improvement that either expands resources or distributes resources more efficiently.
4) Shipping in additional resources to alleviate the effects of "not enough to go around". (Which is a VERY temporary measure, as the population would then expand to where even that amount becomes insufficient.)
My bet is on severe birth control, but developed in such a way that citizens _want_ to have fewer offspring.