Insinuate means you imply or suggest something that may or may not be true. If you say things seemed to go wrong about the time your brother took over, you insinuate that he had something to do with the decline.
There's another way to insinuate. Suppose you're in line to get into a popular dance club when a celebrity appears, surrounded by a big entourage. If you strike up a conversation with one of the entourage, you may be able to insinuate that you're part of the group and go in with them. Don't feel bad — people have been doing it at least since the 1520s, when insinuate evolved from the Latin word insinuare, meaning "wind one's way into."