Do you have a plunge router? If so then you need to get a plunging bit to do the operation.
I would not try to lower a bit into a hole that small, I don't think it would be very safe to do. You might want to think about a template and the use of a smaller dia. bit....
To answer the basic question, with right bit and the right set-up you can do what you are wanting to do. Depending on a lot of factors you are talking about 4 passes to get the depth.
Have you found hardware for attaching the side rails yet? If not you might want to do that before you get too far on the head and foot parts. Some of these parts require slots or other cuts in the post that might change how you cut the mortising for the other boards.
Ed
I would not try to lower a bit into a hole that small, I don't think it would be very safe to do. You might want to think about a template and the use of a smaller dia. bit....
To answer the basic question, with right bit and the right set-up you can do what you are wanting to do. Depending on a lot of factors you are talking about 4 passes to get the depth.
Have you found hardware for attaching the side rails yet? If not you might want to do that before you get too far on the head and foot parts. Some of these parts require slots or other cuts in the post that might change how you cut the mortising for the other boards.
Ed