It wasn't the Titanic that sank on the night of April 14, 1912—it was its identical twin, the Olympic.
While Olympic was being patched up after her encounter with Hawke the solution to the owners' and builder's problems would have became obvious. Even as they cannibalised parts from the second, almost complete and practically identical, sister ship the idea of a complete switch must have occurred. After all, It wouldn't be the first or last time that the identities of ships had been transferred one to another. To this day the most common major maritime fraud is to switch the names of ships. If the ships were switched then the damaged Olympic, masquerading as her sister Titanic, could be lost at sea and an insurance claim made. [Link, via Metafilter]