In fact, they started doing so in just their second year of existence, somehow making the postseason after starting 11-9. Probably the fact that the league, the Basketball Association of America (BAA) only had eight teams in it had something to do with it, but even Boston has been outdone - and handily - since.
To my knowledge, no team has ever dug out of a worse-looking start than the 1996-97 Phoenix Suns, going 0-13 before righting the ship to arrive at a 40-42 record after canning their coach, replaced with Danny Ainge of all people (they would lose in the first round to the Seattle Supersonics).
The 2004-05 Chicago Bulls were nearly as bad, who went winless for their first nine games before turning things around and making the playoffs for the first time since Michael Jordan wore red and white (they would also lose in the first round, to the Washington Wizards).
The 1967-68 Bulls were similarly awful, starting the season 1-15 before reversing their fortunes enough to achieve a gentleman's sweep by the Los Angeles Lakers in the postseason, and the 2003-04 Miami Heat were a solid challenger for worst ugly start after making the Eastern Conference Semis (that they would lose to the Indiana Pacers) after starting 0-7.
Most of Boston's tough starts are pedestrian in comparison, like Danny's first full season in Boston as a GM (2003-04, 8-12 with a loss to the Pacers in the first round), the season after that (9-11, losing again to Indiana in the first round), or Brad Stevens' second year as head coach, when the Cs went 7-13 to open the season, to end up 41-41 (and being swept by the Cleveland Cavaliers in the first round).
The astute reader will note that most (though not all) such teams are exhibiting a disturbing trend contrasted with the lofty aspirations of this current iteration of the Celtics - lots of first-round exits. It's not all doom and gloom in that regard, as the previously-noted Heat managed to not only make the Eastern Conference Semis, they'd pull off a similar feat seven years later, making it to the NBA Finals after a 9-8 start.
The 2010-11 Heat started 9-8, the 2014-14 Cavs were 19-20 at one point. Those teams figured it out. Will the 2017-18 Thunder? They're now 4-6 and just lost to the 1-8 Kings.
Of course, this is the exception, rather than the rule, and we shouldn't assume this kind of outcome just because it's possible. It'll take real work and experimentation to fight back into the form of a legitimate contender, and that challenge lies squarely in the hands of the players. But, conversely, it's not off the table, either, as history has shown us - and with the path as wide as it's been in maybe a decade for a real chance at a ring for the Cs, there's no better time than the present to try.