Question

Is the sum of two polynomials of degree 5 always a polynomial of degree 5

Answer 1

No.

Example
$P(x)=x^5+x+3\\ R(x)=-x^5+3x-4\\\\ P(x)+R(x)=x^5+x+3+(-x^5)+3x-4=4x-1$

Answer 2

The 5th degrees could cancel out to be a polynomial of degree 4. Similarly for lower degrees. 
But it can't be a 6th degree.