That is for sure an overgeneralisation, but so is Wikipedia's. There are some elements of truth in both: Ancient Rome held that freedom could not be sold, and in principle a freeborn person could not become a slave.
[F]reedom was, like servitude, conceptusliased as a natural state. Thus, it was in principle, if not quite in practice, impossible to surrender one's freedom, except in very special cases. As one Roman orator declared, 'What nature gave to the freeborn cannot be snatched away b any injury of fortune.' Roman law thus considered free status inalienable . . . Throughout Rome's history the basic division appears to have remained intact.
Mouritsen, Henrik. The Freedman in the Roman world. Cambridge University Press, 2011.
In practice, many freeborn Romans did become slaves, and still more were functionally enslaved by fellow Romans, if not quite technically slaves.
Despite the popular myth, poor Roman parents could not legally sell their children into real slavery. For instance, a law issued by Constantine in 325 states that even though a father had the power of life and death over his children, he could not sell their liberty (Libertati a maioribus tantum inpensum est, ut patribus, quibus ius vitae in liberos necisque potestas permissa est, eripere libertatem non liceret.).
Yet we have good evidence of Roman parents doing just that from all periods. The Roman state would not sanction the practice in law, but also demurred in putting an end to it.
Cicero [thought] that patria potestas had originally included some kind of a right to sell one's children, but for the period for which we have sources, the selling of freeborn children or pledging them loco servi were understood as offences - but with no punishment to parents.
Ando, Clifford, Paul J. du Plessis, and Kaius Tuori, eds. The Oxford Handbook of Roman Law and Society. Oxford University Press, 2016.
In practice, therefore, children were indeed put to work like slaves in exchange for money to their parents. By law these transactions were legally void and the children retain their freeborn status, but it probably made little practical difference. Nonetheless, unlike manumitted slaves, sold children do not become freedman on redemption but regain their full native freedoms.
Likewise, there were many affirmations that exposed freeborn infants do not become slaves even if reared as such by a rescuer. In an age before DNA testing, however, this was basically theoretical.
Ultimately, at some point, Roman law compromised its dogmatic principle with the prevalent practice, by recognising a parental right to "rent" their children out. According to the Pauli Sententiae:
Whoever, under threat of extreme necessity or for the sake of sustenance, sells their own children, shall not prejudice their freeborn status. For a freeman has no price . . . Parents may nevertheless rent the labor of their children.
Harper, Kyle. Slavery in the Late Roman World, AD 275–425. Cambridge University Press, 2011.
The above source also mentions that in the 420s, Augustine states that the law permits a rental period of 25 years. Again, while theoretically different from slavery, there is little practical difference for the children in question.
Aside from children, it was not generally possible for freeborn Romans to be legally enslaved by private transaction, and those who found themselves in chains could have their freedoms restored by a magistrate. Accordingly, free Romans could not legally sell themselves into slavery. In fact, Roman law recognised such transactions (if the "slave" shared the profits) to be fraud, since their free status could not be given up. Ironically, the punishment was for the perpetrator to be actually enslaved.
Roman law guaranteed the citizen's status to such an extent that the status could be forfeited only by legally recognised processes: capture in war, or when a free man sold himself as a slave in order to defraud an unsuspecting buyer.
Wiedemann, Thomas. Greek and Roman Slavery. Routledge, 2003.
Nonetheless, it was possible for freeborn Romans to be sentenced to slavery, or slave-like conditions. In the High Empire, a punishment known as servi poenae developed, whereby both lower class freeborns and slaves could be sentenced to theoretically lifelong hard labour, typically in a mine. This is known as "penal slavery", and it was a condition even worse than usual slavery.
These 'slaves of the penalty' were neither the property of the fiscus nor the property of the emperor. Nor did servi poenae enjoy the usual rights of a slave. An edict of Antonius Pius appears to have ruled that slaves of the fiscus could receive an inheritance, while servie poenae could not. Moreover, slaves condemned to the metalla and saved by the clemency of the emperor - according to Ulpian - were not returned to their owner but could become servi fisci.
Hirt, Alfred Michael. Imperial mines and quarries in the Roman world: Organizational Aspects, 27 BC-AD 235. Oxford University Press, 2010
Legal punishments and black market transactions aside, another major source of enslaved Romans came from kidnappings. This was, obviously, not supposed to be legal. In practice, pirate kidnappers had no regard for their victim's status and slave traders asked few questions. In an age before personal identification and extensive government records, it would have been quite difficult for a freeborn person, once kidnapped and sold to a distant part of the Empire, to free themselves via the legal process.
With regards to enslavement by tax collectors, there are a few well known examples including the Jews, the Frisii and Bityhnia. It is said that, when Rome requested Bithynian soldiers for the Cimbrian War, king Nicomedes III replied that all his able bodied men had been enslaved by Roman tax collectors. Likewise, the Frisii revolted when the Roman tax burden became so intolerable, that tax collectors enslaved their wives and children. Finally, Josephtus reports that Cassius reduced four Jewish cities to slavery for failing to pay their taxes.
In these commonly cited examples, it seems clear that the victims were Roman "allies" or subjugated nations, rather than actual Romans. I am unable to find evidence that enslavement was a legally authorised punishment for Roman citizen. In any case, during this time Rome typically outsourced its tax collection to local agents, sometimes known as "tax farmers".