At the end of the day conversion is always a choice. Awareness of conversion is not always immediate, but the fruits of conversion are undeniable where people have been converted. We have only to look for them and honestly acknowledge that they are there. The Lord said, "By their fruits ye shall know them".
The word "mistake" means "to take in error".
In general I believe it is not possible to prevent people from willfully "taking in error" the testimony that something is true by their taking it to be false, contrary to their own knowledge.
However, what is manifest from the Holy Ghost and the Light of Christ is self-evident.
Behold, here is the agency of man, and here is the condemnation of man; because that which was from the beginning is plainly manifest unto them, and they receive not the light.
And every man whose spirit receiveth not the light is under condemnation. (Doctrine and Covenants 93:31-32, denominational teachings of the Church of Jesus Christ of Latter-day Saints)
People can receive a testimony but subsequently neglect it or fail to appreciate it for what it is--just like we can breathe air and not be grateful nor consider that God has done it. But it is there and all who know what it is must either give God the glory, or diminish themselves through ingratitude.
People can choose to ignore it.
People can misinterpret it.
People can misrepresent it.
But they cannot deny it without misrepresenting what they know.
So there is possibility for "mistakes" on our end. But in the end, all will stand to account before God for how we held the truth that was revealed to us, and how we would have treated the truths that could have been revealed to us.
What is revealed of God is self-evident, but we do not always recognize that evidence.
Numbers 12 does appear to show there are degrees of apparentness in the revelations that God gives:
If there be a prophet among you, I the Lord will make myself known unto him in a vision, and will speak unto him in a dream.
My servant Moses is not so, who is faithful in all mine house.
With him will I speak mouth to mouth [face to face], even apparently, and not in dark speeches; and the similitude of the Lord shall he behold: wherefore then were ye not afraid to speak against my servant Moses? (Numbers 12:6-8)
However, this does not in any way diminish from the reality that God speaks to us and gives us knowledge in His still, small voice. If we are prone, or if we are not already aware of the existence and nature of God, there is a way we can be awakened to the reality of these things.
He sends His servants telling people about the reality of God, the purpose of life, and how to prepare for eternity through repentance and the redemption brought about by the Son of God.
It is necessary that
that the time shall come when the knowledge of a Savior shall spread throughout every nation, kindred, tongue, and people. (Mosiah 3:20)
It is ultimately impossible to remain neutral. We must choose what portion of the Savior's teachings to accept, whether it be none, some, or all of them. This will determine our degree of happiness in eternity.
What God has planted in our hearts, we cannot deny. We actually know it. How then could we lack constant awareness of it? An example shows that a person may not be aware of what is happening in the moment of his conversion:
And whoso cometh unto me with a broken heart and a contrite spirit, him will I baptize with fire and with the Holy Ghost, even as the Lamanites, because of their faith in me at the time of their conversion, were baptized with fire and with the Holy Ghost, and they knew it not. (3 Nephi 9:20)
Some have mistakenly thought they did not have a testimony, but they did. In one example, brother Heber J. Grant said he believed the Gospel but did not know it was true. In response the prophet told his counselor,
[Heber] knows it just as well as you do. The only thing that he does not know is that he does know it.” (Knowing that we Know, Elder Douglas Callister)
In the Church of Jesus Christ of Latter-day Saints we are admonished and covenant by the weekly ordinance of the sacrament to "Always remember Him".
Interestingly the Lord prefers to let people remain in ignorance or fail in their awareness, including of their identity as children of God, rather than have them knowingly and unpreparedly rebel against or fail to live up to that knowledge of their identity. This life is the time to prepare to meet God; we are not yet prepared, and must prepare quickly.
This might seem like a contradiction in terms but it is not; it is the way He works to allow people as much goodness as they are willing to receive. We have been granted a space in which to repent, after which the end comes, and all will be judged according to their works and the desires of their hearts.
For of him unto whom much is given much is required; and he who sins against the greater light shall receive the greater condemnation. (Doctrine and Covenants 82:3)
This is why we must actually prepare ourselves to receive truth. It is not child's play. God will require us to increase what we are given, which if we do, we shall "enter ... into the joy of [our] Lord".