Combined impulse responses
You can readily use the associativity of the convolution to explain this fact. Assume you send a signal $x(t)$ through a chain of convolutions $h_i(t)$:

$$ y(t) = (((x(t)*h_1(t))*h_2(t))*h_3(t)) $$

By associativity, this becomes

$$ y(t) = x(t) * ((h_1(t)*h_2(t))*h_3(t)) $$

I.e. you can see this operation as if sending your signal $x(t)$ through a single filter with impulse response $h(t)=h_1(t)*h_2(t)*h_3(t)$.