Depends on the Conspiracy and depends on the reasoning behind it, depends on the information involved. Discussing conspiracies and fringe beliefs in detail is a prospect that you can't do proper justice to in a single post, but take ANY conspiracy and look at the evidence and you can usually find and understand the logic behind why people believe in it and why when you set the government or common analysis beside it you find flaws and begin to consider that it is possible. Take two of the biggest fringe beliefs in the US (because I'm an American, I'll go with that since I don't want to do any other nationalities an insult by discussing your own unusual incidents).
UFO crash at Roswell, and JFK Assassination. Pretty much everyone has heard about these two no matter where you're from.
The UFO Crash at Roswell.
According to the accounts, an extraterrestrial spacecraft crashed between Roswell and Corona New Mexico in 1947. The common fringe belief is that a craft complete with bodies was recovered by the US government and it was concealed. There is massive amounts of evidence that says it was a combination of events, partly a project Mogul Balloon, a Top Secret project to detect Nuclear Weapons tests, partly Air Force parachute dummy testing, and a general cover up of one Top Secret Project. The skeptics cite numerous inconsistencies that mainly stem from the eyewitness statements of individuals that were there who claimed to have seen craft, bodies, debris and otherwise. A number of them were military personnel who should have known the difference between a 'flying saucer' and a balloon...even eye witness statements who claimed to see four foot tall aliens and the government says they saw six foot tall parachute dummies. Lost records that were supposed to have been kept indefinitely (Roswell Army Air Field base logs and outgoing reports for 1946-1948). Materials that were supposedly vastly different from those of a Mogul Balloon being reported at the crash. Add in Personnel reports that the Chief of staff of the Air-force made a specific detour to Roswell at the same time...it makes you wonder.
Look at the two and you see a consistent thread, individuals say one thing, and people are told 'you didn't see that, you're completely wrong'. It's not so much what is being said as who is saying it that causes some issue. First hand accounts (by individuals who were there, and there is an exceedingly long list of those accounts that rarely get discussed) and second hand accounts from Military personnel gives a large body of statements that say one thing, while the government says something else that contradicts it. You're talking about (supposedly) honest people saying something, and experts who have an advanced (for the time) knowledge base who are all saying 'this wasn't human'. And the government is saying 'you're all wrong, nothing to see here you squashed events from over six years into one memory'.
A plausible issue exists with the official account, and eyewitnesses can support it.
The JFK Assassination.
Setting aside the eye witness statements, the probable cause on the part of everyone from the Soviets to the Mob to Fidel Castro to the US Army. I'm going to cite one reason that people believe that there was a conspiracy;
The US Government said there was.
Of the two groups that investigated the Kennedy Assassination, (The Warren Commission and the House Select Committee on Assassinations) the first one claimed that Lee Harvey Oswald was a lone gunman operating alone who killed Kennedy. According to the House Select Committee on Assassinations, JFK was most likely killed by two gunmen, with four shots being fired, one being fired from the Grassy Knoll.
Now setting aside eyewitness testimony of other gunmen, setting aside inconsistent information on the Autopsy of President Kennedy at Bethesda and the suspicious circumstances around it, setting aside the dynamic of Jack Ruby and other parties involved in the whole affair, once again, you have the US Government, who has access to all the information all the most up to date and cutting edge technology, operating with unlimited resources and earnestly searching for the truth...come up with two separate conclusions. Which one are you to believe? Which one can you work off of?
Now yes, the Acoustic evidence that the HSCoA decided represented a second gunman on the Grassy knoll was challenged in 2001 and 2003. But then you have the witness testimony that says shots came from the Knoll. Again, there in lies the problem...a plausible issue exists and a body of evidence supports it.
It's somewhat disingenuous to simply say that 'conspiracies are only believed by nutjobs' because the big ones tend to stem not from outlandish grasping at straws, but by plausible issues in the main report that have a body of evidence and testimony supporting it. Yes, there are conspiracies out there that are inane and ludicrous which are only continued by the misinformed such as the arguments that 9/11 was an inside job or that Roosevelt knew about the Japanese Attack on Pearl Harbor before the fact.
