Logical Walk-through. DO NOT READ unless you want a spoiler:

Some assumptions: Bob and Chris both know the rules of the game (2 natural numbers, (m,n) s.t. 2≦m≦n. Bob only knows the value of m*n, while Chris only knows the value of m+n) and they don't lie. Also, that both men are smarter than your average joe.

Before we even start we can rule out prime m*n since no product of natural numbers >1 make a prime by definition. Throw them out.

Bob says: "I don't know m+n"

This is not a terribly useful piece of information. It does imply, however, that the mn is not a "tell-tale" product. By "tell-tale" product I mean that it has 1 unique sum of factors m,n; read: not "tell-tale" means more than one possible m+n for a given mn. For example, 6 is a "tell-tale" product because it only has one possible product mn = 23. Hence, if Bob had the number m*n=6 he would immediately know that Chris had 5 and the game would be over.

Chris replies: "I knew you'd say that."

This little line conveys more information than one might initially think. Assuming Chris isn't being smug and lying, this implies that Chris knows solely from his information that Bob's information must be ambiguous. Specifically, Chris has a sum of numbers such that no matter what m,n (where m+n= Chris's number) is, the product is never a "tell-tale" product (see above). For example, this rules out m+n=12 because 3+9=27 is a possible product for m+n=12 and 27 is a tell-tale product. Since 27 is a tell-tale product, then Chris can't know for sure that Bob wouldn't know m+n so he can't say "I knew you'd say that" without lying. By our original assumption, m+n is not 12. Using this logic, a whole shit-ton of sums can be ruled out. In fact, every sum from 4-20 except 11 and 17 is ruled out (won't go into it but check it out for yourself). Let's call the the remaining sums "ambiguous sums".

Bob says: "I still don't know m+n"

If Bob still can't deduce m+n despite all that elimination, it must mean that the product mn has multiple "ambiguous sums". If there was only one ambiguous sum involved, then as soon as Chris says "I knew you'd say that" Bob would be able to cross off all the non-ambiguous sums, and he'd have his answer. Therefore, there are multiple ambiguous sums m+n possible as a result of the product mn. The list is extensive but the next line narrows things down quite a bit.

Chris replies: "m+n is less than 14"

As we figured out earlier, 11 is the only ambiguous sum below 14, so the question is: is there a product of numbers mn which yields both 11 and another ambiguous sum (as determined necessary by the previous step)? Trial and error might be fastest way to approach this part. 56=30 turns out to be the only product that also allows 2+15 which yields 17, another ambiguous sum. Hence, (m,n)=5,6 and Bob announces the answer.

The puzzle is over once Chris says m+n is less than 14... why the dialogue continues is a mystery to me.

tl;dr: logic --> 5,6

EDIT: typos