Are Measurements made on the same patient independent? I am currently working on the classification of pulmonary diseases using spirometry. This is a procedure in which the patient blows air in a tube and we collect air volume, pressure, etc, in order to obtain  the spirometric parameters. 
My question is : If I perform the spirometry in the same patient three times, can I consider these three exams, at the same day,  to be three different data points in my training of testing set or it is better to average the results and consider only one data point? If the patient comes in a different day, can I considered this exams to be independent?
I think it is okay to consider the exams to be different data points, but I would like to hear other opinions.
 A: I mostly concur with @PeterFlom's answer. In my opinion, you should not average your data (you are basically throwing away 2/3 of your information, why would you want to do that?), but you should definitely account for the fact that measurements on the same patient will tend to be closer together than measurements on different patients. In such a situation, I usually recommend mixed linear models, which are a simple instance of the multi-level models @PeterFlom recommends.
Specifically, you would use a generalized linear mixed model. The link function would be logistic, as in "ordinary" logistic regression. However, the functional form would include multiple observations on each participant, modeled by a random effect, just as in "ordinary" linear mixed models, $y∼F(Xβ+Zγ)$. In R, you can fit this by glmer() in the lme4 package, using the binomial family. For prediction, you could use a single measurement.
Whether or not a mixed model predicts better than a non-mixed model in a particular setting is hard to say, of course. What the mixed model does is account for intra-person variability. If you just average the three original data points, you lose all the variability between measurements, so you will be too optimistic about your ability to predict from a single new observation.
If, on the other hand, you simply throw in all observations without taking the grouping into account, you will again be too optimistic, as all standard errors will shrink. Think of what would happen if you started with a single observation per participant, say 100 data points... and then simply copied each observation 100 times. You would end up with 10,000 "observations" and far smaller standard errors than with the original data, although you didn't enter any new information.
In addition, mixed models allow modeling other grouping factors, like the location, its specific demographics, its staff, diagnostician characteristics etc. So they are a lot more general than averaging.
A: The three exams are different data points. Though they are clearly not independent (nor random) observations of all possible exams in your population of interest, at least for any analysis I can imagine.
Others have emphasized that you may do well to include those data points in your analysis (since you already have them), as simple replicates within patient [a nested design] or including "time/visit" as an absolute (e.g. date) or relative (number of visit) variable of interest [some form of repeated-measures design], if interesting. I agree that this is the most interesting (and probable) scenario.
However, it may not be necessary, pay for increased complexity, or improve your conclusions if you are only interested in between-subjects variables. Let's say that you only care for differences between males and females, or you want to explain air volume by patient age. Since you know that you can not properly characterize a patient in one blow (cause measurements result variable even for the same patient at the same moment), then you take several measures and average them. You don't care about that variation, it's just inevitable; you just want to get as close as possible to the "true" (mean) value for that patient (at/in that time). This may be the most reasonable analysis.
[Check this paper for a good read about simplicity vs. complexity in statistical analyses.]
A: They are definitely three different data points, but they are also definitely not independent (whether they are same day or different day). What you should do about this depends on the goals of your analysis, but it is likely that a multi-level model is a good choice. Averaging the points is also possible, but it reduces variability and eliminates the ability to look at trends over time. 
A: In accordance with the other answers (no, these observations are certainly not independent, so what do you do about it)....
But do you want to use this information to predict other variables? Many of the suggestions so far seem to be assuming you want to use spirometry as a dependent variable, and thus modelling the error is more straightforward (using a multilevel model). If you instead want to use the spirometry measures as an independent variable, you would be well served by using a confirmatory factor analysis model with the 3 repeat measures modeled as indicators of a single underlying latent variable. The variance of the underlying latent variable is that shared by all three measures, and thus a better reflection of what you are really after (compared to taking the mean, for example). 
A: the measurements can be independent or not. if you describe the measured value as $y_t=x_t+\varepsilon_t$, where $x_t$ - true value, and $\varepsilon_t$ - measurement error, then independence means that $cov(\varepsilon_t,\varepsilon_{t-i})=0$ for all times. this may or may not be true. if you have two measurements one immediately after another then it's most likely not true. if two measurements were time separated but conducted buy the same technician again this may not be true. etc.
on the other hand it must be possible to setup the measurement in a way that $\varepsilon_t$ would be independent of each other and the $x_t$. 
$y_t$'s are most definitely not independent through $x_t$ correlations, but that's not what is meant by independence
