deg(pq)=deg(p)+deg(q) 
After following this theorem, I am curious as to when:
$deg(pq) < deg(p) + deg(q)$
Can anyone give me either an example when this is the case or conditions for this to be true?

When the product of their leading coefficients is 0 (for polynomials in one variable). If there is more than one such coefficient (for multivariate polynomials), then the sum of their pairwise products must be 0.
Example: Ring $\left( {{\mathbb{Z}_6}, + , \cdot } \right)$ and polynomial ring $\left( {{\mathbb{Z}_6}, + , \cdot } \right)\left[ X \right]$. $\left( {3{x^2} + 1} \right)\left( {2{x^3} + 1} \right) = 2{x^3} + 3{x^2} + 1$
Example 2: In $\left( {{\mathbb{Z}_6}, + , \cdot } \right)\left[ {X,Y} \right]$, $\left( {2{x^2}y + 2x{y^2} + 1} \right)\left( {3{x^3} + 3{y^3} + 1} \right) = 0\left( {{x^5}y + {x^2}{y^4} + {x^4}{y^2} + x{y^5}} \right) + 2{x^2}y + 2x{y^2} + 3{x^3} + 3{y^3} + 1$
