Just a guess but the posted signs may be for private property that may exist between the road and the stream. So the stream and its immediate banks are a trout management area but maybe a lot of land bordering the stream banks is private. Or sometimes people post land that they don't own or have the permission of the owner to post just to keep people out for whatever reason. See if the signs are signed with a name of a person or organization that can be tracked down for verification. If they are not signed they I would simply ignore them and go fishing.
Trout Unlimited has some literature on this stream and where to access for legal fishing.

http://www.nutmegtrout.org/fishing/where-to-go/aspetuck-river/

Here is the DEEP trout stocking map with stocking locations. Do a copy/paste into your address line and hit enter.

http://www.ct.gov/deep/lib/deep/fishing/stockingmaps/aspetuck_river,_weston.pdf