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In my textbook there is the following definition:

A sequence of random variables $\{X_n\}$ is uniformly bounded if there exists $K>0$ such that $P(|X_n|\ge K)=0$ for all $n$.

There are no examples given, so am I right saying that a sequence of normal random variables, $X_n\sim N(\mu_n, \sigma_n)$, will not be uniformly bounded, but a sequence of Bernoulli random variables, $X_n\sim B(p_n)$, or Binomial random variables, $X_m\sim Bin(n_m, p_m)$ will be if the parameters of the distributions are fixed?

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You are correct, a normal random variable has density $e^{-(x-\mu)^2/2\sigma^2}$ which does not have compact support, hence $P(X\le x)>0$ for all $x\in\Bbb R$. On the other hand, if you have a sequence of Bernoulli random variables, parameter fixed or not does not matter, you know they take values in the set $\{0,1\}$, so let $K=2$ no matter what $p$ is $P(X<-2)=P(X>2)=0$. Binomial random variables are sums of Bernoulli random variables, so if you have $n$ bounded, then the answer is yes as it will be supported on $[-n,n]$ so $P(X<-n-1)=P(X>n+1)=0,\;$ independent of $p$. The converse is also true.

In the case of any discrete random variable on $\Bbb N$ you can easily see that it is bounded as an individual random variable iff its generating function is a polynomial, so a sequence of discrete, random variables is uniformly bounded iff the degrees of their generating functions are finite and uniformly bounded.

This all comes from the more general fact that a random variable (resp. sequence of random variables) is bounded (resp. uniformly bounded) iff it has compact support (resp. are all supported on a common compact set). In the case it is discrete, you use that discrete + compact implies finite to get the second paragraph I mention.

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    Thank you. This helps a lot. – 2017-02-08
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    @user_0_1_2 my pleasure. Since you're new a pro-tip: be sure to upvote helpful answers you get, and accept ones that give you what you're looking for. It helps future users to know what you found helpful and in searching out good information. Welcome to MSE! – 2017-02-08
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    It says that "votes cast by those with less than 15 reputation are recorded but do not change the publicly displayed post score". – 2017-02-18