Are sentence fragments ever acceptable?
Yes, when used deliberately. A fragment is a group of words punctuated as a sentence without a complete subject-verb structure, and the difference between a stylistic device and an error is whether the writer meant it.
Used deliberately, fragments do specific things.
Emphasis. We tried everything. Every single thing. The fragment lands harder than a clause would.
Pace. A short fragment after long sentences changes the rhythm and creates a pause. Advertising and journalism use this constantly.
Answering a question. Why did it fail? Poor planning.
Dialogue and interior voice, where people do not speak in complete sentences.
They become errors when accidental — usually when a subordinate clause is punctuated as though it were independent:
Because the deadline moved. Which was unexpected. Running late as usual.
Each of those needs attaching to the sentence it belongs with, and the giveaway is a leading because, which, although, since or while on something standing alone.
The practical distinction: a deliberate fragment is usually short, and it makes complete sense to the reader because the missing information sits in the previous sentence. An accidental one leaves the reader waiting for something that never arrives.
Context matters too. Fragments suit journalism, marketing, fiction and conversational writing. Academic, legal and technical writing generally expect complete sentences throughout, and a fragment there reads as a slip regardless of intent.
If you use them, use them sparingly. A page of fragments loses the contrast that makes them effective and simply reads as choppy.
A quick test if you are unsure whether one is deliberate: read the fragment aloud on its own, with no surrounding text. If it still communicates something, it is probably working. If it sounds like a half-finished thought, it is an accident.