@ Rolf
Sorry, but it is not likely I can do much about this. I suspect that it is the search character limit your web host has placed on your database. Most web hosts seem to set a 3 character limit for “shortwords”. Shortwords are ignored when searching the database. My guess is that your web host has this set to 4 characters instead of the typical 3 character limit. This means any word that is 4 characters or less will be ignored by the database search.
Connections does have logic in place to deal with shortwords, but only if those words are between 2–3 characters long. So, when you are searching, the 4 characters are too long to be dealt with by Connections and its shortword logic while being too short that they are discarded by the database when doing the search.
I do have to ask… how truly impactful is this? It does not seem all that likely that someone will search for less than half a name.