It is good that the focus of the question is the biblical basis, and not biblical verses, for (despite what some Christians might think, or claim) there is not even one verse in the entire Bible that could be rightly used to support the idea of the Bible being in error. This is the view of Reformed Protestants who still stick to their credal statements on the topic. Some Protestant groups, however, have moved away from this fundamental stance.
What is done by some who are critical of the Bible in that respect, is to take bits - various accounts - and subject them to higher criticism, which has its scholars then decree that those are in error (for example, the Bart Ehrman school which has many supporters today, some of whom say they are Christians). That could be historic events, or miracles, or just a few words wrongly translated, here and there. It needs to be pointed out, though, that the doctrine of biblical inerrancy only ever has applied to the autographs - the original manuscripts, and not copies of them. But this brings us to a closely related point; the manuscripts used.
A truly critical point is that the most controversial matter here is not how the text has been translated so much as which texts are translated.
Reformed Protestants believe not only in the divine inerrancy of scripture, but in the divine preservation of scripture. Here is one example:
"The Old Testament in Hebrew... and the New Testament in Greek...
being immediately inspired by God, and by his singular care and
providence kept pure in all ages, are therefore authentical." The
Westminster Confession of Faith (1.8), 1646
The like stance is stated in the Savoy Declaration (1658), the Helvetic Consensus Formula (1675), and the London Baptist Confession (1689).
But, starting in the late 1800s and continuing to this day, other manuscripts not used by the Church from the first century till the 20th, began to be preferred over all that the saints had preserved and promoted, translating from those ancient Hebrew and koine Greek manuscripts that had come down to them, into modern languages so that the populace could read the inspired word of God for themselves. An example of the subtlety of sneaking those in is with the New King James Version, which claims (in its preface) that the O.T. is a translation from the Hebrew Masoretic text, and the N.T. is translated from the same texts used by the A.V. of 1611. Not so when the NKJ margins are examined, where many deviations are shown to have been slipped in, using other modern texts from the Critical text. Many subtle points of difference have been missed by many, as detailed in a 15-page booklet produced by the Trinitarian Bible Society, examining the claims of the NKJ:
"...the character and testimony in our churches will radically
change... for the worse... The Authorised Version is far superior, and
while not perfect it remains the best and most accurate English
translation of God's Holy Word." Critique of the NKJ Version, p.14,
Trinitarian Bible Society, 2008 See also their article "The Twin
Doctrines of Scripture" in their Quarterly Record, Issue No. 624,
July-September 2018, pp.39-40
Their gloomy prediction has proved true. We now find those who use the NKJV supposing it's the same as the AV, just without thee, thou, and strange words like 'shambles' for the market-place. Not so. Worse, teachings based on subtle distortions in the Critical Text have corrupted the relationship of Christ and God, with some feeling so liberated by the Critical Text that they can now teach false doctrines, and get off with it, almost unchallenged.
My five-fold answer is that today's disbelief in the doctrine of biblical inerrancy arises from (1) disbelieving God would preserve his inspired scriptures throughout the centuries; (2) that the comparatively recent discovery of supposedly better manuscripts (because of being older) has caused those to replace what was "once for all delivered to the saints" (Jude verse 3) in the first century; (3) that modern translations based on those have given Higher Critics a field-day because of the many conflicting translations we now have; (4) that availability of many modern versions has perversely caused many Christians to be less well-read in scripture, even to being almost biblically illiterate, happily being spoon-fed by those who claim to be 'qualified' teachers, the drip-drip influence of the Higher Critics hastening their 'take-it'or-leave-it' attitude to the Bible. (5) a corrupted biblical text is the main basis for disbelief in the inerrancy of scripture, for it gives rise to error.
I base that on over 40 years as a Protestant Christian, mixing in various denominations, where even my own one has largely succumbed to using modern versions based on the Critical Text. Truly we see the fulfillment of the scriptural warning, that...
"...the days come, saith the Lord God, that I will send a famine in
the land, not a famine of bread, nor a thirst for water, but of
hearing the words of the Lord. And they shall wander from sea to sea,
and from the north even to the east, they shall run to and fro to seek
the word of the Lord, and shall not find it." Amos 8:11-12 A.V.