@pursuant,
It really is not recommended,as there will be severe drops in performance and stability for the last extender. Placing this many wireless hops in your network could also cause potential problems with dropped connections and addressing.
This is the gap that mesh networks fill. For some REs and Routers, a mesh can be created through either Onemesh or EasyMesh features, however, the RE220 does not have these features, and it requires a compatible router to create and manage the "mesh" nodes.
It's worth mentioning, as its a common mistake with range extenders, that you are ideally looking to place a Range Extender halfway between your router and the area in which you would like to extend the network too. Placing the RE in a room with a weak signal will cause the RE to suffer from the exact same experience that you have with your other devices.
Try placing the RE outside of the bedroom or in a hallway, as this will allow the RE to receive a stable enough connection to perform decently. If you are able to, you may wire your existing RE220 to your router, which allows the RE to operate as an Access Point, and then you MAY be able to extend your network sufficiently