Is it possible to believe (incorrectly) that one is on the path?

Yes and ... yes.

One can be wrong about the foundation, the method one chooses, one can be wrong about anything! This is why it is important to study the Buddhadharma first and understand its foundational teachings (any introductory book will do) and to continue to study it outside of practice sessions. Practice is very important because there is a lot you will sort out through the extremely simple practice and heart of Buddhism: vipassana meditation aka self-observation.

The thing I believe you are talking about though is can one be wrong about one's own experiences? One can do this, dangerously generating what Master Huai-Chin Nan calls "false gong-fu", things that you learn from somewhere and you want that experience so strongly that your subconscious mind (which is very powerful in creating entire experiences such as dreams) replicates the experience, oftentimes as a precursor and obstacle to the real phenomenon.

For example, this can happen with kundalini in your body, chi rotations, mental phenomena, and even great peace and ease that occur when you practice Buddhist meditation (or any meditation) correctly. (See the end of the first chapter of Tao & Longevity of Master Huai-Chin Nan for excellent discussion of this.)

This is why the Buddha never talked about chi, chakras, and various things. All of those things he labelled "fire element" (which is absolutely correct). Thus, his monks did not create mental obstacles for themselves and rapidly progressed to the heart of understanding of non-self and interdependent origination.

That is the answer to the first question. As for your second question, a master would say "no, you are wasting your time, you can lose your life and opportunity to Awaken any day, any hour.. why waste your breath with false phenomena?"

And that master would be right by saying "no" to your second question.

Nonetheless, I would append my own answer: I would argue though that some of these false phenomena, although they can confuse and betray a person for years and make a person lose decades of lifetime... they can serve as very specific lessons about oneself in the future towards Buddhahood cultivation, (whether one is almost a bodhisattva or whether one is already a bodhisattva and working towards Buddhahood) where the smallest hairline of an error can cause drastic errors. This lesson-about-lessons has a thousand Buddhist parallels and parables (e.g. cosmological time in being stuck in the higher absorptions, wrongs when committed in higher realms, deadly result of losing your essence when in a higher stage of physiological cultivation).

This is similar to how overcoming small past failures (e.g. overeating, drug addiction, tv addiction, etc.) can give you the edge you need (discipline, focus, sorrow, memory) to succeed in later challenges, the failing of which can end your career or cripple or kill you.

So yeah, if you stumbled over something and ran in circles, contemplate the lesson you learned, be grateful you had the lesson to learn something valuable about yourself for such a relatively small cost, write an essay, swear to reform and move forward!

Absolutely, it is possible to be incorrect about the notion that one was on the Buddhist path. I have had plenty of this sort of experience in my own meditation practice.

Very frequently I would have unusual experiences in meditation, such as seeing lights, being filled with joy, a sense of time coming to a stop, etc. And I would ask my teacher what these experiences meant, hoping for this kind of answer or that kind of answer. Each time, he would tell me that these experiences don't mean anything and that I should focus on seeing these experiences as they are.

While I wanted these experiences to return quite badly, I failed to recognize the fact that I wanted them to return. Entire meditation sessions were wasted waiting for the lights, or time stopping, etc. It is in this sense that I "thought I was on the path" but now know that I was somewhat stubbornly misleading myself. Well, try and try again.

Although this example is from practice in a Theravadan context, the general principle should still translate to Mahayana practice or any kind of practiced skill: Sometimes it takes a lot of mistakes to realize you need a different approach. This is definitely the case for me.