It depend of your random number generator algorythm.

see attachment.

I took 2x512 random numbers, added them up and took the average as you said. Sorted the resulting numbers and created a chart out of it. If the line would be linear it would be a linear distribution. From what I can rad out of it it really is a bell distribution.

Hm, let's throw two dice.

result     probability
1          0
2          1/36
3          2/36
4          3/36
5          4/36
6          5/36
7          6/36
8          5/36
9          4/36
10         3/36
11         2/36
12         1/36

Now wouldn't that be a triangle? You need more dice to get the bell. One die gives a line, two dice give a triangle, more dice twist the triangle towards the bell.

It will approach a bell shaped if the initial distribution is flat.
Sketch of proof: near the top of the distribution, the number of ways to create the maximum value remains the same (i.e. 1). The same holds for the minimum value you can generate.
However, near the middle of the distribution, there are more ways to generate the result (you can write down the number of permutations that yields the same result, but I can't be bothered). You can even go further and take the limit of N measurements (instead of 2) and show that the shape of the distribution tends to the normal distribution. You can test this easily with a couple of dice.

Punchline: if you sum two distributions, the result is a new and different distribution (a sum of flat distributions is no longer flat, nor is the a of normal distributions normal).

Now I'm confused too!

Yes. You should divide it if you want to keep the same normalisation.

To clarify what I said in my first post:
The sum of N flat distributions tends to a normal distribution as N tends to infinity.

True, but it's easier to compare the distributions if they're both normalised.

http://en.wikipedia.org/wiki/Normal_distribution

Thanks.