Some molecules containing polar covalent bonds are nonpolar. Explain how this is possible?

You stated "Molecules that contain polar covalent bonds typically have regions of positive and negative chage and thus are polar". This is true, but the polarity that you're referring to is a physical property of the molecule as a whole and not the individual bonds that make up the molecule. Compounds that are considered non-polar may have electronegative and electropositive elements - It's the overall dipole moment of the entire molecule that counts. Take dichloromethane, CH2Cl2, for example. Chlorine is much more electronegative than carbon so a dipole moment must exist, yet the molecule is considered to be non-polar. The reason is because the molecule is symmetrical and, therefore, the dipole moments  at each carbon center cancel each other out. Chloroform (CHCl3), on the other hand, lacks this symmetry and is, in fact, polar...

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