No matter how many observations are made, they do not prove a generalisation. This can be seen with the example of the inductivist turkey:

This turkey found that, on his first morning at the turkey farm, he was fed at 9 a.m. However, being a good inductivist, he did not jump to conclusions. He waited until he had collected a large number of observations of the fact that he was fed at 9 a.m., and he made these observations under a wide variety of circumstances, on Wednesdays and Thursdays, on warm days and cold days, on rainy days and dry days. Each day, he added another observation statement to his list. Finally, his inductivist conscience was satisfied and he carried out an inductive inference to conclude, "I am always fed at 9 a.m.". Alas, this conclusion was shown to be false in no uncertain manner when, on Christmas eve, instead of being fed, he had his throat cut. An inductive inference with true premises has led to a false conclusion. (Chalmers, p. 14)

Similarly scientists cannot prove their hypotheses merely through testing them. Consider the logic of the following:

1. If the hypothesis, H, is correct then the empirical event E will be observed.
2. The event, E, is observed.
3. Therefore hypothesis, H, is correct.

No. 3. does not necessarily follow from the other two. However the following argument is considered to be valid.

1. If the hypothesis, H, is correct then the empirical event E will be observed.
2. The event, E, is not observed.
3. Therefore hypothesis, H, is not correct.

This means that it is always possible to disprove a theory but never possible to prove a scientific theory. Science progresses through falsification of old theories rather than proving of new ones.

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Barry Barnes, About Science, Basil Blackwell, Oxford, 1985, chapter 2.

Alan Chalmers, What is this thing called Science, 2nd edition, University of Queensland Press, St. Lucia, 1982.

Penelope Sanderson, Inductivism, its problems, and the theory dependence of observation, 1999.