Power of summed signals
The formula you gave is an estimator for the &quot;statistical power&quot;, i.e. the expectation of the square of a continuous  underlying random process:
$$ P_X = E(X^2(t)) $$
Computing the latter theoretically would require to know the probabilistic behavior of $x(t)$, which you don't have access to in real life, especially since:

you get sampled data (not continuous),
you generally have a limited number of realizations.

This is even more complicated when you add variables, because you should know their paired probability behavior. A classical result is that, for variables $X$ and $Y$,
$$\mathrm{Var}(aX+bY) = a^2\mathrm{Var}(X)+ b^2\mathrm{Var}(Y)+ 2.a.b.\mathrm{Cov}(X,Y)$$
This quantity can take different values, depending on the last term: $\mathrm{Cov}(X,Y)$. For uncorrelated variables, the covariance vanishes, and the variables add as:
$$\mathrm{Var}(aX+bY) = a^2\mathrm{Var}(X)+ b^2\mathrm{Var}(Y)$$
So, for uncorrelated signals, powers add. And this generalizes to several signals.
Note that when  you sum over a time frame to estimate such quantities, you somehow assumes that you can estimate quantities by averaging in time, with assume that the signals are correctly sampled for a stationary and ergodic process.
For $K$ signals of  identical power $P$, the combined power can take value:

minimum: $0$ when signals sum to zero,
$KP$ when uncorrelated,
maximum: $K^2 P$ when signals are identical.

Additional pointers:

Addition of signal variance and noise variance
Variance of Uncorrelated Variables
Signal averaging
Superposing Signals