Can nonpolar molecules have polar bonds?

Sometimes, there can be.

If there is, it must all cancel out. If you consider #"CO"_2#, we know that's nonpolar linear:

#:stackrel(..)"O"="C"=stackrel(..)"O":#

but we should note that it has two dipole moment vectors in the #z# direction:

#stackrel(larr)( :stackrel(..)"O"=)"C"stackrel(rarr)(=stackrel(..)"O": )#

These are of equal magnitude but opposite direction, so they cancel out perfectly. Thus, #"CO"_2# is considered nonpolar.

However, it doesn't mean #"CO"_2# can't be polarized, which is another story. It has two infrared bands near #"2360 cm"^(-1)#.

These correspond to #"CO"_2#'s asymmetrical stretching and bending vibrational modes. It has two nondegenerate modes that are IR-active as shown below:

If it were not possible for #"CO"_2# to develop a net nonzero dipole moment (thereby changing its dipole moment, one prerequisite for an IR band, the other being the asymmetry of the vibrational mode), we wouldn't see these infrared bands.

We could have treated these identical-magnitude dipole moment vectors as #vecl = << 0,0, 1 >># and #vecr = << 0,0,-1 >>#. In that case, we can show that they cancel upon summing them together:

#vecl + vecr#

#= << 0,0,1 >> + << 0,0,-1 >>#

#= << 0,0,0 >> = vec0#

which is a dot, representing the nonpolar result of two dipole moment vectors of identical magnitude and perfectly opposite directions summing together.