I'm running a 2X2 between-subjects design ANCOVA and wonder about one peculiar thing: The ANCOVA output table reveals no significant interaction effect, albeit the profile plot shows two lines that cross each other, i.e. an disordinal interaction. So how to reconcile these apparently contradictory findings? Might I presume the existence of an interaction effect (which would make a lot of sense by the way..) based on the plot despite a lack of statistical significance? And why is it non-significant; perhaps because of a lack of statistical power?
There's no necessary contradiction.
The crossing can be present but not significantly different from zero. Lines that don't cross are not necessarily 'weaker' in effect than lines that do.
Lack of a significance may not just be a lack of power; it might be that there's actually no population effect (or at least none of any practical import - i.e. none that we'd care to identify).
That said, there could well be a 'real' interaction (one large enough to be worth finding) and, as you suggest, a lack of power (relative to the size of the random variation in the data) is why you didn't find it.