If there could be a counter-argument against a premise, does that make the
No. The mere possibility of a counter-argument (i.e., "there could be a counter-argument") doesn't imply that the premise is false or that an argument containing the premise is unsound. The counter-argument itself must have a true conclusion in order to guarantee that the premise against which it's a counter-argument is false.
Every sound argument has a true conclusion (although the converse doesn't hold), so if there exists a sound argument against a particular premise, then the particular premise is false. Often, however, the very soundness of that counter-argument will be a matter of controversy.