It has never been fully explained why the Bugaled Breizh trawler from France's Brittany region sank off of Britain's Lizard Point on 15 January 2004.
Families of the five victims believe it was dragged by a submarine, because traces of titanium, which is often used in the hulls of submarines, were found on the ship's cables.
An initial investigation in 2008 found that submarines of various Nato countries were performing drills in the area.
But following the assessments of two export reports, which determined there was no submarine operating where the ship went down, French authorities ruled out the theory in 2013.
French authorities have said the fishing boat's cables likely got caught on the seabed.
The case was closed in May, but the families launched what one of their lawyers called a "make or break" appeal in the city of Rennes on Tuesday.
Lawyers want permission to interview crew members of submarines that were in the area and to consider further expert reviews.
Dominique Launay, head of a group supporting the families, said they "cannot leave without knowing the truth".
The court can either confirm last May's decision to close the case or order further investigations.