If one says he is Jewish because he converted, is he to be believed?
Marrying a Jew: If a person who was known to be a gentile claims that he went through conversion and became Jewish, he is not believed to be allowed to marry a Jew unless he supplies witnesses [or other proof of his conversion] or reconverts. This applies even if this person now acts like an observant Jew. If however the person was always held to be Jewish and now says that in truth he converted, then he is believed without witnesses. Now although some Poskim argue that even such a person must bring witnesses, nevertheless, the custom is like the former opinion. [In Israel, the Rabanut requires all foreigners to prove their Jewish identity and provide authorized conversion papers in order to be allowed to marry.]
Other matters, such as Tefillin: The above requirement for a gentile to bring proof of his conversion even if he is now an observant Jew applies only with regards to allowing him to marry a Jew. However, regarding all other matters of Judaism [such as Tefillin, Minyan etc], if he acts as an observant Jew he is believed to have converted even if he does not bring proof of his conversion.