Is Poisson distribution assumption appropriate for modeling "students arriving at a local bar"? If we want to model the number of students arriving at a local bar within an hour, is Poisson distribution appropriate for this random variable?
This is an example of Poisson distribution from a textbook, but I think it's not appropriate because a Poisson distribution assumes "the number of events occurring in non-overlapping intervals are independent". I think knowing the number of students arriving in the first few hours tells us something about how busy the bar will be in the next hour. For example, if we see a lot of students in the first few hours, we can predict that today is probably a holiday and there will be more students arriving in the next hour. Therefore  it's wrong to assume that "the number of events occurring in non-overlapping intervals are independent".
Is my reasoning correct?
 A: I think what you're missing here is conditional dependence, and what it is conditioned on.
Suppose we have some Poisson distribution of known mean parameter $\mu$. Then any two samples drawn from this distribution, $n_1$ and $n_2$, are conditionally independent of each other, i.e.
$$
\begin{split}
P(n_1 \mid \mu, n_2) &= P(n_1 \mid \mu), \\
P(n_2 \mid \mu, n_1) &= P(n_2 \mid \mu).
\end{split}
$$
This is true as long as $\mu$ is known. Intuitively, if we know $\mu$, then observing $n_1$ doesn't give us any new information about what we will observe in $n_2$, because we have all the information we need already about the distribution from which it is drawn.
However, you are describing the bar situation as if $\mu$ is unknown. You say,

I think knowing the number of students arriving in the first few hours tells us something about how busy the bar will be in the next hour.

Translation:

Observing $n_1$ tells us something about the value of $\mu$.

Suppose $\mu$ is unknown. Then we can assume some prior, $P(\mu)$, and we can update this prior upon an observation using a Bayesian update rule: $P(\mu \mid n_1) \ne P(\mu).$ Then $n_2$ is not conditionally independent of $n_1$, because before observation of $n_1$, we have,
$$
P(n_2) = \int_0^\infty P(n_2 \mid \mu) P(\mu) d \mu,
$$
and after observing $n_1$,
$$
P(n_2 \mid n_1) = \int_0^\infty P(n_2 \mid \mu) P(\mu \mid n_1) d \mu \ne P(n_2).
$$
Thus, for unknown Poisson parameter $\mu$, $n_2$ is conditionally dependent on $n_1$. However, for known $\mu$, they are conditionally independent. This is probably what your source means when it says that multiple draws from a Poisson distribution are independent. Intuitively, in your example, you are dealing with a Poisson distribution of unknown mean, and you are using an initial sample to hint you towards what the mean may be. However, if you knew $\mu$ in the first place, observing a time-interval sample would grant you no new information.
Now, there may be other reasons why the students coming into a bar in non-overlapping time intervals may not follow a Poisson distribution. What comes to my mind is time-dependence, or people being put off from full bars. But I'm sure for the sake of an example problem, it's reasonable to approximate it as a Poisson distribution.
A: 
For example, if we see a lot of students in the first few hours, we can predict that today is probably a holiday and there will be more students arriving in the next hour. Therefore it's wrong to assume that "the number of events occurring in non-overlapping intervals are independent".

Even if you have information about the number of students in prior hours, the number of events occurring in subsequent hours can still be independent. The information about the first few hours provides an estimate of the mean underlying rate of the Poisson process, for example if it's a holiday with a high average rate. But unless the students themselves are choosing to come into the bar based on such information, each student is still making an independent choice of whether and when to arrive. It's just that the probability of coming into the bar is high for all students, independently of each other, on that particular day.
In practice, there are lots of issues in modeling this type of situation. If students use Google to see how crowded the bar is to decide whether to come, the events clearly aren't independent so a Poisson model would't do. Even with a Poisson process, the underlying rate can change during the course of a day so that there is an inhomogeneous Poisson process. If students typically come in groups, a Poisson model assuming independence won't work. But simply having information about the underlying rate doesn't by itself rule out independent events in a Poisson process.
