From 1592 to 1599, Sigismund III Vasa, king of Poland-Lithuania, seized the Swedish throne, establishing a personal union between the two nations. Sigismund was overthrown, partially due to his insistence on establishing Catholic institutions in Sweden.
Around the same time period, from the 1570s to 1650s, several attempts were made to establish a union with Russia/Muscovy, although none ever succeeded. Sigismund III also made a bid for the Russian tsardom, but Boris Godunov won out.
Is it possible for these unions to all succeed, creating a Poland-Lithuania-Sweden-Russia? If so, what impacts would it have on European (and Asian, or even American) politics?