Are there verbs that are neither telic, or atelic?

These examples from wikipedia are shown as tests to find what is telic, or atelic.

Fine: "John built a house in a month."
Bad: *"John built a house for a month."

    → built a house is telic

Bad: *"John built houses in a month."
Fine: "John built houses for a month."

I'm wondering if there are verb phrases that fit neither. I've been trying to think of what would be a verb like that, but I'm having no success.

I'm also having trouble figuring this out by reading the article. https://en.wikipedia.org/wiki/Telicity

A related question What makes progressive verbs different from atelic, and can a verb be both atelic/telic, and/or progressive?

It can be difficult to determine the telicity of a verb, but all verbs must be telic or atelic in context.

Imagining a verb that is neither telic nor atelic is rather like imagining a binary number that is neither 1 nor 0, or imagining an object that is neither tea nor not tea. Because a telic aspect indicates that an action or event is complete in some sense, and an atelic verb is simply the inverse (an action or event that is incomplete), you're essentially asking if an action or event can be neither complete nor incomplete. I don't think that's possible, as even an indeterminate or ambiguous action/event would have to be one or the other.

However, because telicity is partly determined by context (where/how/by whom the sentence is said) and semantics (what the sentence means), the test posed above is flawed for a verb that can take either expression in context. In other words, because the aspect of telicity depends on the noun phrases following the verb, adverbials, tense, and other contextual features, there are verb forms that can be interpreted as telic or atelic in the same expression. Take the verb eat in the following form:

I ate.

Telic test:

I ate in an hour. (Telic - the action was completed in an hour; clearer with a count direct object like the cake but still telic here)

Atelic test:

I ate for an hour. (Atelic - the action went on for an hour without necessarily being completed; clearer with a noncount direct object like cake but still atelic here)

Fun alternatives involve adding other expressions or intonations to mean "I already ate," "I just ate," "I ate for a while," and other items to create telic or atelic meanings.

In other words, telicity in English isn't as bounded a grammatical aspect as perfect or progressive aspects. It is not only determined at the verb level or even at the verb phrase level, but also by the semantic interactions between verbs and everything around it.

Nonetheless, every use of a verb is ultimately telic or atelic, in the sense that it will be interpreted to describe either a completed action/event or one still in process.

My main scholarly source for this is Hana Filip in Aspect, Eventuality Types, and Nominal Reference (1999). Filip urges moving away from thinking of telicity as an aspect determined on the lexical or phrasal level. While much of the argument is complex, her key claim comes through here:

Aspect shift is traditionally accounted for at the lexical level, in terms of lexical ambiguity. In section 3.8.2, it has been argued that a uniform treatment of aspect shift in terms of lexical rule strategy is not viable, given that lexical rules are not flexible enough to handle the whole range of phenomena that fall under aspect shift.

Recent accounts of meaning shifts, discussed above, treat the dependency between the verb and the adjunct as encoded by some specification either only on the verb or only on the adjunct. There is yet another way to analyze this dependency: namely, verbs (along with their arguments) and adjuncts are mutually constraining, and therefore the relevant constraints could be stated over both of them.

...

The difference between such pairs of sentences indicates that it is not an adjunct on its own that determines the telicity of a given sentence. It is the inherent lexical semantics of the verb together with the semantics of the adjunct that together determine whether a sentence will have a telic or an atelic reading.

If you understand telic to be a description of the utterance in terms of whether it presents a completed act, then the following are telic utterances:

John built a house.

John built the house in only a month.

but depending on what the speaker means to say in context, this could be atelic:

John was building that house for a year (i.e. when he gave up because he ran out of funds).

or telic:

John was building that house for a year (it took him so long because he did it alone).

There's nothing morphological there that indicates the difference.

But one could say that "building that house for a year" is atelic in both cases and that the context in the second presents the atelic utterance in light of the now completed act. John was "at it" for a full year in both cases. In the first he ceased (with the house incomplete) and in the second he ceased (with the house completed).

So you might think of atelic verbs as those verbs or verb phrases (including any adverbial modifiers and direct objects) whose action can cease in media res.

She ate the cracker in one bite. ^telic

She was nibbling at the cracker. ^atelic

She nibbled the cracker.