If two quantum systems are entangled their measurements become correlated, even if they are separated by a distance.
But what if one has two different quantum systems at hand, initially un-entangled and separated, can one generate entanglement between them? Or is it necessary that they be "close enough"? If there is no such theoretical necessity has any experiment ever been performed to achieve this?
Usually it is impossible to entangle spatially separated systems $A$ and $B$, but you can evade this restriction by a trick called entanglement swapping. You entangle $A$ with a third system $C,$ and $B$ with a fourth system $D$. These operations must be performed in the usual way, by allowing $A$ $(B)$ to interact with $C$ $(D)$. Now $C$ and $D$ are brought close together and projectively measured in an entangled basis (i.e., the outcome of the measurement is always that $C$ and $D$ are in one of a set of entangled states). This measurement simultaneously projects $A$ and $B$ into an entangled state (which entangled state depends on the specific outcome of the $CD$ measurement).
Note that $A$ and $B$ can remain arbitrarily far from each other throughout the entire procedure. The entanglement between $A$ and $C$, and between $B$ and $D$ is "swapped" to $A$ and $B$, hence the term. You can think of $C$ and $D$ as representatives for $A$ and $B$, taking their places at the meeting.
Here's a news article citing the first experimental demonstration: https://phys.org/news/2007-10-entanglement-swapping-quantum.html
To further explain the prohibition on entangling distant systems, the actual statement is that entanglement between $A$ and $B$ does not increase under local operations and classical communication (LOCC). In entanglement swapping we have nonclassical communication (transmission of quantum states $C$ and $D$), and also nonlocal operations (measurement of the joint state of $C$ and $D$), so that's how the trick works.
Entanglement is synonymous to ignorance!
In general, it arises whenever trying to break down a quantum system into subsystems. In more mathematical terms of quantum mechanics, this means to decompose the Hilbert space $\mathcal{H}$ of the full system into a Hilbert space $\mathcal{H}_{A}$ of a subsystem $A$ and a Hilbert space $\mathcal{H}_{\bar{A}}$ of the complimentary subsystem $\bar{A}$. As a result, given a pure state $|\Psi\rangle\in\mathcal{H}$ of the full system, it can be written in terms of pure states $|\psi^{A}_{i}\rangle\in\mathcal{H}_{A}$ and $|\psi^{\bar{A}}_{\bar{i}}\rangle\in\mathcal{H}_{\bar{A}}$ of the subsystems as a mixed state,
\begin{equation} |\Psi\rangle = \sum_{i,\bar{i}} c_{i\bar{i}} |\psi^{A}_{i}\rangle \otimes |\psi^{\bar{A}}_{\bar{i}}\rangle \end{equation}
Entanglement is present whenever $|c_{i\bar{i}}|\ne1$, $\forall i,\bar{i}$. To put it into perspective, the fact that the entire universe cannot be measured at the same time forces us to consider subsystems that are more controllable. However, this procedure of breaking down the system comes with unavoidable ignorance of the true nature of the full system, i.e. entanglement.
So, to answer your question, initially un-entangled, i.e. pure, states cannot become entangled because this would invoke an information loss paradox since the initial pure states that are equipped with "deterministic" information suddenly become mixed states whose information content has reduced due to ignorance. To give you more understanding of your own inquiry, you are asking the same question that caused the black hole war between Leonard Susskind and Stephen Hawking. After Hawking discovered his Hawking radiation, the information loss paradox came into play: "Throw a pure state in the black hole and wait long enough to obtain the Hawking radiation which describes now a thermal, mixed state. Where did the information go!?"
