Can a collection of random processes be not random? A friend and I were having a debate about randomness and at one point, I said that it was possible to have a collection of random processes which were not random when "put together." He disagreed. 
So, I put the question here more concretely and with more detail. 

Suppose I have a large number of random processes, is it possible for the collection of those processes to non-random and also, is it possible to have a part of that collection be not random? 

Thanks for the help

You are correct. The key is using dependence. For example, let $X$ be distributed as a continuous uniform random variable on the interval $[0,1].$ Let $Y=1-X.$ 
Then define $Z=X+Y.$ Now $Z$ is a constant, but composed of two random components. 
There are more practical examples. Imagine a closed-loop system where components move among several states randomly. The sum of all components is fixed and non-random, but the number in each state is a random variable. You can also have one or more states that are not random, satisfying your last version. 
