Yes, a (strongly) suspected infringer would be aware of the (at mininum) suspicision of infringement well before a suit would be filed, IMHO.

You have to consider the process....

In the USA, the patent owner must identify suspected infringers through whatever means (on hardware-type items, this would typically be through a tear down and analysis of the infringer's infringing product, as an example). The patent owner must then provide formal Notice to the suspected infringer (and this starts the clock for the term of infringement - legally), along with data demonstrating why their product is suspected of/accused of infringment.

This Notice would typically invite contact and negotiation. Only when negotiations fail would the patent owner - typically - file suit (as a last resort).

So, to again answer your question, IMHO the accused infringer will typically have been very aware of the suspected infringement (and be presented an offer to negotiate settlement) well ahead of an formal legal action. It's inherent in the process.

I am not an attorney, but this is my understanding of the typical patent prosecution process. However, I'm sure there are many instances where the patent owner opts to file suit concurrent with providing Notice to an infringer, thus blind-siding. But IMHO such action would be considered quite "rude and crude" in the business community (and would not be helpful, most likely, in any future negotiation), and would also be frowned upon by the Court. Again, I'm not an attorney, but it wouldn't surprise me if a legal prerequisite to filing suit would be a requirement to have at least pursued negotiation if only to demonstrate that there is a real, documented, legal dispute.

FWIW,

SGE