Can a nonpolar molecule contain polar covalent bonds?
Given the nature of a polar molecule having a net dipole moment in any one direction (think CHCl3 where the net dipole is in the direction of the three chlorine atoms), one doesn’t need to know a whole lot about chemistry to find an example.
Thinking about molecular geometries, electronegativity, and atoms that can become hypervalent, AsF6 comes to mind. Arsenic is capable of becoming hypervalent (having more than 8 valence electrons), it forms an octahedral molecular geometry (more on this in a bit), and the bonds are considered polar given the |△χ|=|χAs−χF| is greater than or equal to 2.0.
Note: the symbol χ is used to denote electronegativity, some textbooks will use EN instead. They mean the same thing.
Now, given the octahedral molecular geometry and the polar bonds, each F-As bond has a dipole moment toward the fluorine, but given the geometry, each fluorine has another As-F dipole moment in the opposite direction that negates it. This results in a non-polar molecule with polar bonds.