Can every type of linear filter be modelled by a convolution?
No. It's only LTI (Linear and Time-Invariant) systems that can be modeled with convolution through a unique single impulse response.

For example the systems

$$ y(t) = g(t) x(t) $$ or
$$ y[n] = \sum_{k=0}^{k &lt; n} x[n-k] $$

are both linear but not time-invariant and their output $y[n]$ cannot be computed with the convolution operation ( $\star$ denoting convolution)

$$ y[n] = x[n] \star h[n]$$

as such an impulse response $h[n]$ does not exist.