I found this interesting information about different correlations. An example is that, the number of people drowned by falling into a pool correlates to number of films Nicolas Cage appeared in. This shows that not all correlation are reliable. My question now is, how do you know if a correlation is reliable or not? How do you know if the correlation makes sense?
You need to be aware of two important issues when considering the correlation between two variables. (1) Spurious results and (2) nature of causation.
Spurious correlation refers to correlation that is false, often created by other unaccounted factors. Famous example, the number of bird sightings indeed correlates with the number of babies being born (possibly resulting in the false and rather ironic hypothesis, that storks bring our babies!) A good read here is the seminal paper by Phillips "Understanding Spurious Regressions in Econometrics". Knowledge of the pitfalls of time-series econometrics, in particular, using differenced series to obtain stationary variables or regressions that employ co-integration can alleviate the spurious nature of correlations.
Secondly, it may be the case the a correlation is spurious as a result of some unknown third variable having a relationship with the first two variables. This is also known as confounding. The famous example here is that people that drink coffee, have a higher probability of dying from cancer. The spurious relationship comes obviously from the fact that people that like to drink coffee, also smoke more cigarettes. Another interesting, but more rare, example is that a correlation can only exits if another variable is accounted for. For example, knowledge of nutrition may help you to keep your weight; but you need money to buy good food and maintain your diet in the long run. So, money here established the correlation between knowledge of nutrition and weight, although by itself the simple correlation may be not existing. To control for omitted factors and extract the true relationship between the variables, we either need to control for other factors to obtain the so called 'partial' correlation or we use advanced techniques, such as random trial experiments. Moreover, when dealing with secondary data, we use 2SLS and instrumental variable methods to effectively purge the relationship from confounding factors.
Causation is another problem, and more of philosophical nature. What causes what? In case of cigarettes and cancer the causation might be clear. But what about income democracy and corruption? Are states with higher corruption less democratic or are more democratic states less corrupt? If our logic fails, 2SLS methodology can alleviate the issue. With regards to time series, causality tests (Granger, Sims) have been developed, but they may not be very robust and based on time lags instead of true causation.
To sum it up, the first test is logic. Then, look at experimental results under a controlled study and/or apply econometric/statistical tools to confirm your hypothesis about a correlation. The better your identification strategy, the more robust your results will appear; however, crystal-clear correlation, as it is the case in natural sciences, is difficult to establish in social or behavioral sciences.