What are the examples of 'not a sentence'?

Answer 1

A sentence is not a sentence if it lacks a subject or a verb. Examples:

Up until then...

...as often as possible.

The car with the dent...

...if you can. (However, without the "if", the statement "You can." is a sentence.)

After lunch...

...of a sentence.

It should be noted that when people are talking to each other, it is not as necessary to always use complete or proper sentences; it is important when writing.

Answer 2

Fragments, which do not express a complete thought, are examples of 'not a sentence.' For instance, "walking to the store" is a fragment because it lacks a subject and a verb. Another example is "because I can," which is a dependent clause and does not form a complete sentence on its own.