The degree of the sum is the larger of the two starting degrees. I am doing a course in algebra, in the course professor is teaching " The degree of the sum is the larger of the two starting degrees" which means if we add two polynomial having the same degree then after doing the sum the total degree will add up. I am finding the flaw in the above quoted statement. How come the degree of the sum will be larger it has to be same. 
If we add $x^2 + x^2$ the sum will be $2x^2$, degree remains same.
 A: If it's not just a poor and unfortunate translation (your profile doesn't say where you come from, so I don't know if that's a possibility), it appears as if you have misunderstood your professor. 
There's a difference between "larger than" and "larger of".
The degree of the sum of two polynomials will always be equal to or smaller than the larger of the degrees of the addends, in most cases, e.g. if the two addends does not have the same degree, it will be equal. There are examples in the comments and in gimusi's answer of it becoming smaller, the thing being that the two terms of highest degree cancel each other because their coefficients add to zero.
As in got mentioned in a comment: If we're dealing with multiplication, the two terms of the highest degree (let's say $ax^b$ and $cx^d$) get multiplied and the result has degree $b+d$ and the coefficient is $ac$ which is not zero so the degree of the result is always $b+d$, i.e. the sum of the degrees. 
A: No we have that the degree of the sum is less than or equal to the maximum of the two starting degrees, let us consider as an example 
$$(x^2+2x+1)+(-x^2+3)=2x+4$$
