The answer is unfortunately "it depends". It depends on the airport layout, the specific flight route, and the rules and regulations of the frequent flyer program and your specific access tool.
It's a lot easier when you can stay airside. If you are flying domestic in the US you do arrive airside and you can chose when to leave the secure area and pick up your bags. In this case you can just walk to the local lounge and check whether they will let you in.
If you have to go ground side it's more difficult. Some airlines have a lounge ground side but this is rare and typically not easy to get in. You can ask for a "gate pass". That's equivalent to a boarding pass and will get you through security even if you have no flight booked. Some United Club members actually do this: ask for a gate to access the lounge and have a few free drinks and then leave again. See http://www.flyertalk.com/forum/united-airlines-mileageplus/1353861-gate-pass-united-club-access.html
Whether they will grant you access and/or give you a gate pass depends on a large number of factors: if you bought a full fare first class ticket directly from the air line and you are also a high status member the answer will most likely be "yes". If you are trying to get in with an Economy ticket and a status match from a partner airlines, the answer is more likely to be "no".
There are quite a few qualification for access (fare class, airline of booking, credit card, paid membership, partner match, frequent flyer status, etc.) and the rules will all be different which makes it hard to predict