How can a molecule have polar covalent bonds but remain non-polar?

Though the bonds may be polar covalent, the entire molecule
may be symmetrical and hence, non polar

An example may be CH4

The C-H bonds themselves are clearly polar, yet
the symmetry of the molecule means that  there
are no polar 'ends'; an approach of this molecule
by a different polar molecule finds no preferred 'end'
to be attracted to, the entire CH4 molecule is non polar.

Hope that helps! :-)

Basically if you have a molecule that has polar bonds, but each of those polar bonds "cancel out" each other, then the molecule (as a whole) is considered non-polar.

Take a look at CCl4 (carbon tetrachloride). It has 4 C-Cl bonds which are all polar covalent bonds (where Cl is more electronegative than C, so Cl is partially negative and C partially positive) but because the molecule is tetrahedral, there is no net dipole moment in the molecule. Another way of thinking about it is CCl4 is symmetrical, so the electron attractive force on one C-Cl bond cancels out its effect with the C-Cl bond on the opposite side. If you draw a lewis dot diagram of CCl4 that may help explain it better.