Funny because from the same website you get the opposite answer -
http://disneyparksmomspanel.disney.go.com/questions.aspx?pgm=2&pid=46&cid=85&qid=176049
"hat's a great question. You should be able to do exactly that at most Quick Service locations, but not all. However, it is a different story for Table Service restaurants. Unfortunately, generally speaking, the answer is "no" for Table Service restaurants. When my children were younger, we would have to wake them up at every Table Service restaurant. I can't think of a single exception."
Although the Mom's Panel is a great resource, they are not the official word from Disney and sometimes get things wrong.
In fact at the bottom of the webpage is this:
These views are not intended to reflect the opinions of Walt Disney Parks and Resorts Online, its affiliates, its management, its agents or its employees and have not been reviewed or approved by Walt Disney Parks and Resorts Online prior to posting.
OP, the answer is it depends. It depends on the restaurant, how busy it is, how close the tables are etc etc and it can change over time. The best answer is to ask but have a back up (we used a carrier that dd would be transferred to).