not necessarily. it could. but if x and y are truly uncorrelated variables that are observed with unlimited precision, then you still would not reject the null that they are uncorelated

"Truly uncorrelated" is a problematic concept. Most of the time, any two things within each other's light cones, have at least tiny direct or indirect effects on each other. These effects are usually not that tiny in experimental settings because of imperfect experimental tools and procedures.

Like the speed of sound and the height of the table where the experiment is.

Note that the pressure of the air changes with height. For a few inches the change is negligible and very difficult (impossible?) to measure, but the effect must be out there.

I had once a problem with the temperature of the room, you usually ignore it, but it has a bigger effect like a 5% variation.