A common question that Calvinists pose to the Arminian is "If Jesus died for all, then why aren't all saved?" John Owen popularised this argument with his "triple dilemma" or "triple choice" where he said [paraphrased] that "Jesus died for either all sins of all men, all the sins of some men, or some sins of all men."
Some non-Calvinists (in taking Owens third choice) have tried to defend their view by saying that Jesus died for all sins of all men, except for the sin of [continual] unbelief. While I respect their attempts to reconcile the facts that Jesus died for all, and that not all will be saved, I reject way in which they try to harmonise theses two truths.
Universalists (in taking Owens first choice) generally appeal to the same passages that Arminians appeal to in support of unlimited atonement. They however, like the Calvinist make the mistake of thinking that not having all saved and having atonement for all poses some sort of contradiction.
Calvinists (in taking Owens second choice) believe that Jesus died for all the sins of some men. In doing so they jump through many hoops to avoid the implications of the countless passages that Jesus died for everyone, and those passages which say God would rather that those who perish would repent. At the same time they point to verses which say Jesus died for "his people" or "his sheep" etc. (Note that these verses don't exclude the rest of the world. Their logic here is flawed, and is akin to me saying that "I love my mother" and then deducing that I don't love anyone else.)
Arminians, including myself (and also Amyraldians) argue that the atonement is provisional. That is, Jesus died for all, but the benefits of the atonement are applied to those who are in Christ. In fact some Calvinists now accept that the atonement is provisional (although say it is only provided for the elect.) In doing so these Calvinists concede that unlimited atonement doesn't lead to universalism.
But those Calvinists who follow Owen and refuse to admit that the atonement is provisional (whether for all or just the elect) walk into a problem of their own. If they deny that the benefits of the atonement are applied only once a person exercises saving faith (Such Calvinists typically claim "You believe God is your potential saviour, where I believe Jesus actually saved me at the cross.") then they must believe that they were then never under God's wrath (as opposed to scripture e.g. Ephesians 2:1-3). They must believe that they were born saved (as there is apparently nothing remaining that is required to make them a beneficiary of the atonement). So then to these people, faith cannot be the means by which they access grace (Romans 5:1-2), but rather a revelation that they are already saved!