No, it is not always possible to identify (or eliminate) the writer of every sample of handwriting.
Handwriting and signatures can be a powerful and significant indicator of identity. Handwriting is, in fact, one of the best forms of personal identification. However, that does not mean that every sample of writing can be identified or associated to a specific person or suspect. The strength of association (or non-association) depends entirely upon the quality of samples used for the comparison.
A meaningful and effective comparison and evaluation requires a sufficient amount of comparable material (both questioned and specimen in nature) with writing that has been executed in a normal and natural manner while displaying adequate complexity and skill. If any of those conditions are not fully met, then the evaluation and final conclusion is likely to be limited.
For example, some people write their signature or handwriting in a very simplistic style or manner with very little complexity in graphical formation. Alternatively (or at the same time), they may write with poor fluency and execution. Writing done in that manner is relatively easy to simulate or copy. Furthermore, low complexity, or unskilled, writing may also possess limited ‘individuality’ meaning there will be a higher chance of coincidental similarities with the writing of some other person.
These factors, among others, must be taken into account when doing a comparison. The bottom line is that good samples make for a good comparison providing the best chance of success when assessing potential authorship.