To pull an example from The Codex Alera book series, there is a standing law that says the Crown must pay triple the offered bribe to a certain type of high-security guard if the guardsman in question turns in the attempted briber.
So, you might want to research up on any special laws against bribery in whatever city you enter, or find yourself turned in, and someone else made much much richer than you ever could have.
Also, who is the proper person to offer a bribe to? Obviously in the case of simple bribery, it's the person you're talking to, but if you're trying to smuggle something into an imprisoned friend, you probably need to bribe the guard captain to send his soldiers on break for a period of time. Sometimes the grunts just don't have the ability to do what you want to bribe them for.
As a small note, bureaucrats and their red tape tend to be the best people to bribe, as well as the most willing. A country filled with them should be a cakewalk to deal with, if you have enough money.