Can we say, "My sister is sitting by the table, reading a book"?
I'd understand by the table and at the table as two different positions.
Sitting at a table would be a position appropriate for working or eating. Your legs are probably under the table and your forearms and hands are on or just above the table. If you read at a table, the book is probably on the table.
Sitting by a table suggests that you are near or next to the table, but your legs are not under the table and your arms and hands are not directly on or above it. You might be able to lean one of your elbows on the table if you're near enough. If you read by a table, the book might be in your lap.
I see it that way too.
Also, a person by a table doesn't have to be sitting:
The dead body was lying by the table.
The stranger was standing by the table.
(I know this wasn't the context you were asking about, but it's something to be aware of when you use 'at' and 'table' together.)
Thanks for all the replies.