As far as I know, yes, but if they find out you will be fired and they'll probably tell the other company, unless it's McDonald's vs. Burger King, which I doubt in this case...
You can if the other competitor does not know but most companies will let you go when they find out you are working for the competitor because they feel it is conspiracy.
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It depends. Some jobs will ask you to sign an agreement that says you won't work for a competitor... but others know that you will. For instance, if you are teaching adjunct for a college... they know that if they don't give you full-time hours, that you will be out there teaching for other colleges as well. You can't disclose proprietary information, but they aren't silly enough to ask you to work for bad money and no insurance, and then say you can't work for other schools at the same time.
You have to check your employment contract. Many companies with direct competitors in the area will have a clause in the employment contract that will not only prohibit you from working simultaneously with their competitor, but will extend that 6 months or longer. This practice is to prevent the exchange of trade secrets. This is very common in the cable, satellite dish, telephone, and internet companies. It also occurs in the food industry and some retail companies as well. If you try to apply at a competitor, they will usually just deny employment, however if you are caught violating your contract, expect a pretty weighty lawsuit.