This happens more often than one would think. My favorite example were the three classroom teachers in one of my graduate courses (who had complained throughout my course how much they hated the rampant cheating by the students in their own classrooms) that turned in a group webpage assignment to me that they had obviously downloaded off the Internet. Leaving aside how stupid they must think me that I wouldn't already know every relevant website and recognized it immediately, I was greatly amused that they thought they could pass this plagiarism off by the simple expedient of substituting their names on the title page. When confronted, the leader of the team bald-faced told me that this was entirely their work and "that other site must have stolen it from us". The other two just looked at their feet and shuffled a bit, leaving it to their team leader to hang them all. So I used the "page source" tab to show them the HTML in their browser, and scrolled down to the metadata tags which included, "Author: Debbie E. Middleton" and "Creation date: 1999".
Students always seem so amazed that one can spot the material they've downloaded from the net and dropped into the centre of their paper, but of course they have no understanding of the (to me) obvious differences in diction, sentence structure, and thoughtfulness. Most students who cheat lack sufficient knowledge and skills to get away with it; because if they knew enough, they probably wouldn't have bothered cheating in the first place. At least for written work.
The exception here is multiple choice tests where a quick glance to the desk next to them gives them the answer to #24. Higher IQ students cheat less often because they know the odds of getting a correct answer from a lower-performing peer are lower than figuring it out for themselves, but everyone gets panicky sometimes, or has a momentary lapse. But this is an easy one to address: simply print four or five versions of the exam, identical in very way, except the order of the questions / alternatives is reordered so the answer to 21 is A on this paper but c on that one. Being identical in all other respects, no one is disadvantaged. Being a different order, everytime a student cheats by looking at their neighbour's paper, their grade goes down 1 point. The greater the cheating, the lower the mark, with no intervention from the instructor required. As you collect the papers, you sort them into the appropriate pile for version A through D and apply the appropriate answer key. With word processing (or better yet, one of the software programs for online tests) this is ridiculously easy to manage and saves a lot of unpleasantness, recriminations, and (worst of all) false accusations of cheating.
On the whole, though, don't become obsessed with issues of cheating. Unless you have reason to believe there is a serious problem, normal precautions are usually sufficient. No precautions does, unfortunately, actively encourage cheating since it sends the message you don't care. But standard invigilation of exams—no leaving the room, no doing your own work at the front of the room, no asking the secretary to cover for you—sends the message that you and your institution care, but it's nothing personal. Taking excessive precautions sends the message that you do not trust students, which can ultimately undermine classroom learning because participation in classroom discussion requires some level of trust. Instead, take the position that you know students are anxious to find out what they know and can do (i.e., that they wish to complete an authentic assessment). If you really believe that, they will too. Our actions convey our expectations, and students generally live up to our expectations for good or ill, which argues against going overboard in a campaign against cheating.
On the other hand, we owe it to good students to not allow them to be penalized for being honest while all around them cheaters prosper. Middle ground, then, eh?
The best way to decrease cheating is to assign tasks that actually have meaning and matter to the students, that allow students to take some ownership. Ironically, in an attempt to avoid students downloading material from the web, many instructors assign obscure topics that aren't even on Wikipedia to ensure that the student is forced to produce original material. But a topic too obscure to be on the Web is too obscure for any course; a topic so narrow they cannot cheat is too narrow to be meaningful. By increasing student alienation from the assigned task, one increases the likelihood of cheating. (And there is no topic too narrow or obscure that they can't hire someone on the web to write it for them.) Instead, let them loose on what is relevant and authentically useful to them, and students will happily do the work. Occasionally, students will cheat on even a worthwhile assignment if their workload becomes too overwhelming to do it all, but then maybe instructors in your program should get together to talk about student workloads....
When students do cheat on authentic assessments, it's almost always obvious. A colleague, for example, used a journal assignment in his course, and one ESL student downloaded an online diary instead of attempting her own. Of course, it was immediately obvious that the diction was too Canadian to have been written by this ESL student, but when the student denied it, it was a simple matter of point out diary entries that could not possibly apply, given that the student was not of that church, did not grow up in that country, did not in fact match any of the particulars of the diary entries.
So assume no one is a cheater, unless the system in which you are teaching is promoting cheating through careless disinterest. If cheating is a problem where you teach, look first to bad structures and poorly thought-through assignments and then at the administrators who are turning a blind eye; rather than glare at students. Lobby for best practices among colleagues before blaming student populations. Once those administrative and collegial issues have been addressed, the majority of students will not want to cheat, and the (vanishingly small) minority of habitual cheaters may be addressed on a case by case basis. Or, to put it another way, if you are encountering a lot of cheating, the problem is you, not (primarily) the students....