Simple way to look at it: a random baseball pitch is a single event, of a relatively large object. It's very "impulsive", in that a baseball hits with a large impulse, infrequently, which you would expect to obey a Poisson distribution.
Suppose you upped the rate, while reducing the size (so that the momentum or mass flow or something like that is more or less the same), which increases the number of particles. Consider a sand blaster: a whole lot of tiny sand particles come out, but when they individually whack into a surface, they don't make the surface go SLAP, SLAP every time, but rather it's the din of a million microscopic twacks, resulting in a dull hiss. The momentum delivered to the surface is rather constant (of course, a regular sandblaster has a lot of compressed air behind it, in addition to the sand, but hold that thought).
If you remove the sand from the compressed air, so it's just air blowing, the momentum is due entirely to the (extremely tiny, extremely numerous) molecules hitting (not even hitting at this point, rather, piling up against it and sliding away), and the amount of noise very small.
In general, the noise in a random variable goes as 1/sqrt(N) for N particles in the system (whatever that happens to mean).
Examples: A sandblaster delivering 10^6 sand grains/second is louder than a sandblaster delivering only 10^4, but not by 100 times, only sqrt(10^6/10^4) = 10 times. A number of equally powerful white noise sources, added together, has a total amplitude of sqrt(N) times (i.e., the noise per source is
1/sqrt(N)). More generally, the noise of N sources with amplitude a_i is sqrt(SUM(a_i^2)) (sum from i=1 to N), the R^N-vector sum of all independent components.
You may already known all this...
Now, applying this to your case, putting more noisy current sources in parallel (equivalent to noisy voltage sources in series; either way, they add) makes the current more even. Noise is increased by a factor of sqrt(N), but the correlated signal increases by N, so the SNR goes as
1/sqrt(N).
In terms of correlation, the DC term is correlated, while all other frequencies are uncorrelated. So the DC term adds, while the others drop out.
Tim