What is the greatest possible sum of two consecutive integers whose product is less than 400?
Let the two integers be $n$ and $n + 1,$ so $n(n + 1) < 400.$  Then the largest possible value of $n$ will be close to the square root of 400, which is $\sqrt{400} = 20.$  For $n = 19,$ $n(n + 1) = 19 \cdot 20 = 380,$ and for $n = 20,$ $n(n + 1) = 20 \cdot 21 = 420,$ so the largest possible sum of two consecutive integers whose product is less than 400 is $19 + 20 = \boxed{39}.$