Can every type of linear filter be modelled by a convolution? I have an input time series going through a filter that creates another time series as output. 

If I assume in first approximation that my filter is linear, does it necessarily mean that I can model this filter by a convolution? 

In other words is it only certain types of linear filters that can be modelled by a convolution? (LT1?...).

I have never seen any clear statement on this in any publication. It is as obscure and vague as definitions of time invariance, ergodicity, causality etc in filtering textbooks. 

My question comes from wondering if there is any point trying to attempt an already risky deconvolution if I am not even sure that my linear filter is necessarily akin to a convolution. 

Why classes of linear filters are not clearly explained in publications (that I know of) is a mystery to me, as it should perhaps be the first question that arises when trying to model real world phenomenons.
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.