Steve and Cody gave good explanation, but a couple things to add (some of my research is about evolutionary consequence of polyploidy).
Steve, in nature, actually, there are a lot of polyploid species. If we consider ancient polyploids, majority (47-100%) of flowering plants have experienced genome doubling. Even Arabidopsis, which has tiny genome, has experienced polyploidization. When we focus on the recent polyploids (i.e. looking at the chromosome numbers within genera), 34.5% of species are polyploids.
Polyploids are extremely important in plant speciation, but majority of advantage is believed to be from combining two genomes of different species (this is called allopolyploid). You can get a new "feature" by combining strengths of two species. When you simply double the chromosome number of a single species (autopolyploid), there isn't much of STRONG advantage, and the effect could vary among species. Generally, the cell size increases with larger nucleus (where DNA is stored), and this causes larger plants (this is the reason why lots of crops are polyploids). But this is not always the case. In some cases, polyploids grow slower, too.
With regard to synthetic polyploids, as Cody mentioned, they use pretty harsh, mutagenic process. Majority of seedlings are sick, and only a few may survive. The surviving ones may show deformed flowers/leaves etc. Also, internal balance of genes (e.g., epigenetics and dosage balance) are screwed up in newly created synthetic polyploids. After a couple generations, the synthetic polyploids can quickly sort out and re-establish the balance, but early generation synthetic polyploids are generally messed up. So I'm skeptical that they are "generally" more vigorous. This instability is less of an issues with naturally occurring variation in ploidy within species.
BTW, recently, most of us use flow cytometry to determine ploidy. It is much easier than chromosome counting, which requires lots of patience and skill. But the machine is a bit expensive, so it may not be accessible for orchid breeders.
Ray, triploids can produce offspring, but the fertility is very very low.
http://barleyworld.org/sites/default..._sterility.pdf
http://www.vitis-vea.de/admin/volltext/e046305.pdf
In the grape, both female and male fertility of 3x is extremely reduced. But if you do enough crossing, you can get a few offspring. During the meiosis, the chromosome pairing is wacky, and a pollen grain may have unbalanced number of chromosomes (e.g., 2 copies of chromosome 1 and 1 copy of chromosome 2, etc). So the dosage balance among genes are screwed up again, and I'm guessing this is a part of the reason why lots of pollen grains from 3x don't germinate or function well. So 3x X 3x is less likely to succeed than 3x X (even numbered ploidy) since both parents are crappy. The resulting offspring will have "aneuploidy" (abnormal number of chromosomes) in any case when 3x is used as a parent.
Finally, using 'N' in orchid world is a bit weird, since we, scientists, use 'x' to indicate ploidy. 'n' is used to report if your chromosome count is done in sporophyte (=adult plant; 2n) or gamete (= egg/sperm; n). For example, a chromosome count of 2n=4x=40 means that there are total of 40 chromosomes in an adult plant nucleus, and this count is done in sporophyte (2n). If we count it in gametes (e.g. pollen), this is same as n=20 (there are 20 chromosomes in the pollen). 4x part indicates the plant is a tetraploid, so there are 10 unique chromosomes, and there are 4 copies of each chromosome in a cell of an adult plant (40 = 4 x 10).
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