I think it depends on your type of pill, dosage, etc. Usually you're supposed to call your doctor and ask him or her whether you should take another dose. If you don't have a regular doctor or gyno, call some place like planned parenthood.
I would use back up protection if you are planning to have sex just as a safe guard. I got really sick on birth control, too. I couldn't find one that didn't make me sick to my stomach. I think the hormones don't work right with everyone's body. I would ask the Dr. Put you on something with a lower hormone dosage.
Well I have become an expert on birth control pills! I have been taking them because of abnormal bleeding and hormone problems after my son was born.
Anyway I know for sure that in the first two hours of taking the pill, it is the absolute most strongest. It is being digested in your stomache and the estrogen is slowly released into your body so if you threw up because of the pill, thats very normal. That will happen. Also around the late afternoon, 4 or 5, your hormones decrease and that will also make you queezy.
Anyway as far as protection goes, ask your doctor of course, but I would take another one that day. Like around 1 pm if you take them in the morning. But of course you ask your doctor. At my 6 week check up I was bleeding so so heavily I had to take three a day to get it under control and it still took almost 3 months. And I got sick a whole lot!! But like someone else said if you planned on having sex just use something extra just incase:)