This website lists about two dozen three-word phrasal verbs. As I looked through the list, they all seemed "inseparable" to me.
However, have said that, I think there is some wiggle room for a few of these, due to the flexibility of prepositions.
For example, consider talk back to. That form may be the preferred version, but one might see talk back at used instead:
When you're in school, don't talk back at your teacher.
Also, there are times when one of the three words can be cut out altogether, resulting in the use a two-word phrasal verb instead. For example, the website lists check up on as a phrasal verb meaning "examine" or "investigate," with the example:
The boys promised to check up on the condition of the summer house from time to time.
I think one could remove the up and retain the same general meaning:
The boys promised to check on the condition of the summer house from time to time.
As for your example, the preposition could probably changed from on to with, and the result would be a more common way of writing it:
Unfortunately, I wasn't able to keep up with the news while I was away on vacation.
Best Answer
In the vast majority of cases, a "phrasal verb" consists of a very common "base" verb coupled with one or more prepositions/adverbs. Thus, for example, if we take the base verb to look,...
As is often the case, the single-word alternatives are significantly less common than the base element in the phrasal verb versions. Sometimes (look sharp, for example) the phrasal verb is undoubtedly informal, but this isn't always (or even, I suspect, usually) the case.
Non native speakers might well think it's even more difficult to learn all the different combinations above than it would be to learn the single-word forms. But from the native speaker's perspective phrasal verbs usually seem easier, because the "building blocks" are so common and familiar.
The net effect of this is that even when a phrasal verb isn't inherently informal, it often seems more appropriate in formal contexts to use a less common single-word form. Apart from anything else, it gives the impression you have a wider vocabulary, since most people would say the highlighted elements in my list are at most "sub-definitions" of the single word look (coupled with various prepositions/adverbs that are so common they barely even count as "words" in the context of an extensive vocabulary).
Since the principles of "formal" English are primarily inculcated within the academic context, where a wide vocabulary is usually seen as highly desirable, students are encouraged (by teachers) and naturally motivated (for their own advancement) to acquire and demonstrate that wide vocabulary.
I hope this explains why phrasal verbs are less common in formal contexts than one might otherwise have expected. It's purely a personal opinion, but I think there's a long-standing general tendency for real (i.e. - informal spoken) English to create more and more phrasal verbs. In total, the language manages to "say more with less" (by using less words, with more significant ways of joining them together).
My advice to learners would be to favour, rather than avoid phrasal verbs, but just be a bit careful with those which are considered "informal", or even "slang". That's what most native speakers do anyway, and I assume most learners aspire to speak like a native, rather than write like a professor (whose prose style might seem awkward/stilted/opaque even to many native speakers).
TL;DR: Use a phrasal verb wherever you know one that means what you want to convey (unless you know it's inappropriate because it's too informal for the context). You'll sound more like a native speaker.