You don't recall correctly. The number of Justices that comprise the Supreme Court is established not by the Constitution but by federal law, as enacted by Congress:
"The Supreme Court of the United States shall consist of a Chief Justice of the United States and eight associate justices, any six of whom shall constitute a quorum."
28 U.S.C. sec. 1
If this provision were to be changed by Congress to establish new seats on the Court, there is nothing that the current members of the Court could do to prevent any new, confirmed Justices from being seated.