It has often been contended that Jesus' disciples stole the body of Christ from the tomb and then proclaimed him resurrected. This conspiracy theory hypothesis was the earliest recorded polemic and given by the Jewish authorities soon after the Easter events. While it may have seemed plausible in the immediate aftermath of the resurrection, the conspiracy theory hypothesis does not match well with many of the facts of history which emerged later.
Although conspiracies have many known problems - ability to maintain a cover up, keeping people from leaking, etc. - and successful conspiracies are rare in general, the biggest problem with the conspiracy hypothesis for the resurrection of Jesus is its anachronistic nature. It fails to take into account the situation of first century Jews who just lost their supposed Messiah and have no recourse or standing in society. These are not the type of people to attempt to foist a hoax upon the world and found a new religion based on their dead teacher - who, they most likely would have thought at the time - may have been a charlatan himself. After all, Jesus spoke of the coming of the kingdom of God, of the inauguration of a new order, and of being one with God himself, yet wound up being executed like a common criminal. Moreover, as recent scholarship shows, most of the early disciples (and likely conspirators) themselves were executed for their beliefs and gained nothing as a result. It is one thing to die for a something you don't know is a lie, it something entirely different to die for something you know to be a lie - particularly as the inventor of such a lie!
A Brief Look at the Fate of the Apostles
In his recent book, The Fate of the Apostles, Sean McDowell wades through the historical evidence for the apostles by focusing on documentation that occurred within "living memory"; on a timeline, before the end of the second century, but practically defined as the time from the apostles themselves to the deaths of those who had direct contact with the disciples of the apostles. This puts a limit on the evidence available and minimizes the development of legends in the stories of the apostles themselves.
Let's turn and look at some of the key apostles.
Peter
Peter's martyrdom comes primarily from the tradition laid out in the Acts of Peter which states that Peter was executed in Rome by crucifixion during Nero's reign and crucified upside down. The Acts of Peter itself dates from the late second century and placed on the edge of living memory by McDowell, meaning that there is likely some historical value in it, although there is much legendary embellishment. However, we can learn something more about Peter's martyrdom from earlier writings, going back to Peter himself as well as those who associated with him. We know from John's gospel as well as Clement and Ignatius' writings that Peter's martyrdom occurred. John and Tertullian (the latter of which sits just beyond living memory) as well as the aforementioned Acts of Peter give evidence to Peter's crucifixion, and Peter's own writings as well as a myriad of other sources testify to his presence in Rome. So while we may not be able to state with certainty that he was crucified upside down in Rome, we can state that Peter was martyred in Rome by crucifixion for his belief in the resurrection of Jesus.
Paul
Paul makes for one of the most unlikely potential conspirators and, in my opinion, his conversion and evangelism make the conspiracy hypothesis itself so ad hoc as to make the theory absurd when trying to account for him. Regardless, history does show us that Paul did willingly die for his belief in the resurrection of Jesus. Ignatius mentions Paul twice in connection with his Roman martyrdom and Clement uses Paul as a model for perseverance through persecution as a "pillar of the faith" because of his martyrdom. Additionally, Paul's own writings discuss his imprisonment, persecution, and foreshadow his coming martyrdom in Rome. The chronology of Paul's life place his death at the time of Nero's persecution of the Christians in Rome and support the traditional tale. Given the repeated imprisonments, beatings, punishments, and eventual death, it seems beyond implausible that Paul would have given up his success and status to form a conspiracy that required all of this sacrifice with no gain whatsoever.
Thomas
One of the more interesting stories comes from Thomas' life and eventual martyrdom. Thomas is traditionally associated with the Indian subcontinent where he was believed to have traveled in his missionary journeys. A major difficulty emerges immediately when seeking to assess the historicity of Thomas' travels, namely the lack of historical documentation - at least by Western standards - in India. The Portuguese brought historical methods with them upon their arrival in the 16th century whereby no written history existed previously. This does not mean no history or historical tradition existed in India, simply it was not documented as it was by Western standards. Regardless, there exist multiple independent traditions of Thomas ministering in India, and there was documented trade between Rome and India at the time. Additionally, there are old traditions of Christians in India who claim a Thomistic lineage as well as reports of Western missionaries encountering pre-existing, Thomistic Indian Christians during the late 2nd century.
The Acts of Thomas falls outside of living memory, but does contain a tradition providing evidence or his martyrdom. There is good reason to believe that the book - despite its legendary developments - contains a historical core with relations to existing Indian cities and rulers which may indicate an elaboration upon historical events in the life of Thomas. Oral traditions were maintained through the centuries which corroborate some of the key events, namely that Thomas traveled to India, preached the Gospel to the people, and was martyred and buried there. The lateness and circumstantial nature of the evidence for Thomas does reduce the probability we can assign to his missionary work in India, but it still points powerfully in that direction.
Other Apostles
For all of the 14 disciples that McDowell studies, he concludes that we have very strong evidence for three of the apostles martyrdoms (Peter, Paul, and James son of Zebedee), high probability for James the brother of Jesus, and that it's more "more probable than not" that Tomas was martyred. Andrew is "more plausible than not" while seven of the remaining disciples are roughly 50:50 regarding their martyrdom leaving only John the son of Zebedee's martyrdom as "improbable" considering there is no evidence he experienced martyrdom. There are also no traditions or evidence of any of the apostles recanting or denying their faith.
A Key Distinction
Many people have died for their beliefs - there's no denying that. But it bears repeating that many of those who died for what they believed to be true, died with belief in things they received secondhand. This contrasts strongly with the apostles and their deaths, because they died believing what they had themselves seen was true. This does not prove that Jesus was resurrected or that the apostles were correct in their belief, but it does go a long way to debunking conspiracy theories centered around a fabrication by the apostles themselves.