If a result tab has no rows, then Alt+Right and Alt+Left won't change to the other tabs. To reproduce, run something like the following in the query window:
select count(*) from Player; select * from Player where PlayerName = 'Foo'; select count(*) from Player;
The result from the first query will be in front. Alt+Right will then select the second tab. If this has zero rows, then Alt+Right and Alt+Left do nothing. You have to use the mouse to select a different result tab.