I once had to bribe airport security in Manila with a carton of cigarettes to get onto a flight to Bangkok, you can read that full story
here, part one is on the first page of that thread, it was a crazy few days.
-One time in Medellin I was pulled over by machine gun toting police while riding in a taxi leaving a gang infested drug ghetto, the cops do this all of the time, I was likely set up after buying the dope. They searched me and found the goods, a quarter ounce of cocaine and five fat blunts (about $35 jajaja) then proceeded to give me a lecture about how dangerous this was. I was already drinking heavily and sniffing rails all night so was fucked up, I was on my way back to a hotel to party with a buddy and two skanks when this happened. After the lecture which included threats of having me deported, the cop said we can make this problem go away for $35 (100.000 pesos), so I gave him the cash and he told me ok you can go now. As I sat back in the taxi, the cop handed me back all of my drugs, only in Colombia!
-Another time in Colombia our night bus from Cali to Medellin (8hours) was pulled over and my retard non Spanish speaking ginger Canuck buddy didn't have any ID, he left it in Cali as we were going back in a few days. So the cops pulled us off the bus, threatened to detain him (near Manizales, in the middle of nowhere at 3am) and even searched our bags, in which they almost found our weed stash, the cop literally had it in his hand at one point! It was two weeks before Christmas so they were looking for easy cash. They started high and I eventually got them down to $50 or so, fuckers. It took about a half an hour and the people on the bus were looking at us like "WTF are these dick head gringos doing?"
Those were all instances where I just wanted to end the problem quickly and was willing to spend a few bucks to do so, I don't lose sleep over $25-50 which is generally what you'll play in a place like Colombia.
I agree to stash your cash in different locations, cards too (best not to have one on you) because the cops are known to take guys to ATMs.
NEVER let the situation escalate to you going to the police station, it'll only get worse because now you're dealing with superiors who may not be willing to take a bribe or if they are, it'll be much more costly.
Always be mindful of the situation when dealing with cops, if it's one then you're ok to bring up the subject but if it's two or especially more then it could be complicated as some cops don't want to be seen as corrupt in front of their colleagues so if you can separate one of them to mention it, that's better.
I once heard a guy say that a threat to call your embassy can work well, which is probably true for the US guys.