Is it unusual to apply for a degree without finishing your current one? Consider the following situation: a student has started a bachelor's/master's/PhD at a lesser-known university program but feels that he would benefit from a more prestigious institution with better professors, equipment, and learning environment.

He decides to apply for a bachelor's/master's/PhD (the same degree as his previous level) program at top schools, planning to abandon his current program if he is accepted.

Is this unusual or frowned upon? Is the fact that the applicant has already started a program likely to hurt the applicant's chances? (Do the answers to the above questions depend on the current level of the applicant?)
Speaking strictly from the US, this is defined as a being a "transfer student" and is extremely common. How common? Well, it's a federally reported statistic called "Transfer-Out Rate". I am fond of the CollegeResults.org tool, which defines this stat nicely (emphasis mine): 


  The percentage of students who began in the 2006 cohort of first-time,
  full-time, bachelor's or equivalent degree-seeking freshmen at the
  institution and transferred to another school without earning a degree
  at the initial institution. Reporting of transfer data is optional for
  colleges and universities that do not consider preparing students for
  transfer as part of their mission. (IPEDS)


For instance, The University of South Alabama is reported as having about 16% of students transfer out to another institution before receiving a degree. The University at Buffalo (New York) posts about a 20% transfer rate.

So, as the write-up for the statistic suggests, some schools even offer specific degree tracks designed to transfer. In the state of Wisconsin, the entire "technical college" offers such programs. In Florida, these are called "junior colleges" and offer similar tracks.

Even at Universities designed without this as their intent, there is always a percentage of people who transfer in and out at an undergraduate level. As Nate Eldredge points out, this is not nearly so common at higher levels, but it happens often enough and is hardly taboo.

The Upsides

Sometimes it is desirable to plan to transfer, such as getting started at a nearby location while you make preparations/plans to go elsewhere (I did this myself, starting in a tech school and applying to my target University just the next semester once I had family/work preparations in place). Many people spend 2-3 years at such colleges, taking all the classes they can towards their higher target degree (usually called "general degree requirements") - usually because it is more convenient and often vastly cheaper (a person can easily save tens of thousands of dollars in tuition this way). Class sizes are often smaller. The difficulty level of the material varies, but most "introduction" classes look a lot an awful lot alike after a while.

The Downsides

If you are looking to go onto higher degrees, especially to a PhD or competitive Masters program, recommendation letters and research experience (when possible) are highly desirable if not absolutely necessary. If you will only be at an institution for 2 years instead of 4-5, then you have a lot less time to find and build these connections! Surely you can do it, but I know I've personally found it took me about 2 years to find the right connections that seem to have "stuck" and suit me and what I want to do (after figuring out what I actually want to do, naturally). If I had completed most of the degree before coming to my present institution, this might mean I'd have missed out on such opportunities to do some real work with these people.

Most institutions have requirements of "credits done in residence", so you can't just do 4/5ths of a degree at Podunk Clown School Of Higher Learning and transfer into Harvard for a semester and get your diploma. Which is unfortunate, because that sounds like it would make for a great B-movie.

Some administrative complications can come into play too. If you decide to retake a course you did poorly on or failed in a previous semester, the rules about how to do this and whether or not it's possible vary depending on where you took the class. Retaking a class you took originally at School A is not always possible or straight-forward once you are in School B.

How GPA is calculated also differs in this same way, as some institutions only calculate grades earned in residence, others combine them, some don't transfer 0 credit classes (like if you failed a class) and others transfer all grades even if they don't give you credit for the course towards your degree! 

The complexity can be a real head-ache sometimes, but they are generally annoyances to be overcome rather than deal-breakers.

Big Warning

Credits earned at one institution aren't automatically accepted anywhere else! Indeed, at some colleges they offer two versions of a class - one that is likely to transfer, and one that isn't; the one that can transfer costs more but is otherwise the same course! This can be a minefield, so if you want to make plans or consider transferring, talk with your target school's Admissions/Registration/Records and whoever would be in charge of a transcript evaluation first!

This process varies hugely by US State and even between institutions within the state (even when they are State schools in the same system!), but suffice it to say that this is an extremely common problem in America and you have to be careful and plan accordingly. 

Admissions Considerations

Finally, you asked about how this might effect perceptions of you in terms of admissions or otherwise. The most important thing I've seen is that once you've had a semester or two at any college, most colleges no longer put much weight into your high school grades, SAT scores, etc - if they even require them at all. So it is very common that people will not do great in high school, go to a local or otherwise non-competitive college for a few semesters, get really get good grades, and then apply to more competitive institutions. Most Universities understand that proven experience in a college setting is a far better predictor of academic success than any high school transcript or standardized test can possibly be, and then put little weight into anything they have to say. This can backfire on you, of course, if you didn't do well at your last program but did better in the past.