You may know this as geirus l'chumrah. A geirus l'chumrah is when a convert (already converted) is required to undergo a "new" conversion because of doubt about a prior conversion. Sometimes this is necessary. That's why it exists. You discover later than a member of your beit din was breaking Shabbos or otherwise became a questionable dayan (judge) at the time of your conversion, you may want a geirus l'chumrah. However, it is not required.
Definition: A geirus l'chumrah requires a new beit din oral examination and a new dip in the mikvah. But this time, there is no bracha said because we generally omit a bracha when the bracha's necessity is questionable. I don't know whether a "new" milah (drop of blood taken from the circumcision area of men) is required.
Here is a link answering the question at the beginning of this post: Questioning the Status of a Halachic Conversion is Anti-Halachic and Unethical.
I wholeheartedly agree with Rabbi Yuter. The geirus l'chumrah process is very painful for many converts, usually is based on nothing more than dislike of another subgroup of orthodoxy, and justifies everyone else questioning the convert (and all other converts). A convert is a Jew, and a Jew is a Jew is a Jew. This is unnecessary and against halacha. It is taking on chumrahs for the purpose of being superior to other people, rather than for the purpose of pleasing Hashem.
And regardless of whether an individual is ever told they must undergo a geirus l'chumrah (it's not exactly a "suggestion"), all orthodox converts today fear that possibility. That possibility is becoming more and more statistically likely today. Worse, today, people now seem to believe that a conversion can be "overturned" because most average Jews have never heard of the term geirus lechumrah before. Because they don't understand the concept, they believe that the convert has been "re-converted" and was not Jewish before. Thus, now the convert's children may be questioned, and they may even be required to do a geirus lechumrah too. These born-Jewish children are also now subject to the sometimes-painful limits that converts must face.
For a stereotypical look at the argument that people "requiring" geirus l'chumrah are not Jews at all, check out this firey blog discussion from 2009: Subbotnik Jews of Ilyinka are Jews. "If these people need GIUR LECHUMRA it means that they are NOT Jews, period, no ifs ands or buts, no matter how emotional and sad their tale..."