Portfolio volatility is not the average of the parts. For two positions it is sqrt(w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 s1 s2 r), where w is weight, s is volatility and r is correlation.
The correlation term is the entire point. Two positions each at 20% annualised volatility, equally weighted: at r = 0 the portfolio sits at 14.1%; at r = 0.5 it is 17.3%; at r = 1 it is the full 20%. Diversification is not owning more things, it is owning things whose correlation term is small - and that term is the first thing to move toward 1 in a selloff.
Measured over a rolling window it becomes a live risk gauge. A book whose volatility has doubled while position count stayed flat has quietly doubled its risk through correlation and volatility regime, and needs sizing cut to hold volatility-targeting constant.
Related: correlation-matrix, volatility-targeting, effective-number-of-bets, diversification