No, as far as I can tell NASA got it right. Voyager 1 isn't going fast enough to reach AC +79 3888's current position in 40,000 years -- but the star is coming to meet it.
Assuming the current speed is 17 km/sec (as stated in the first paragraph if this Wikipedia article, and that it won't slow down much beyond that, it should cover approximately 1 light-year every 17,500 years.
AC +79 3888, also known as Gliese 445, is currently about 17.6 light-years away. But the star itself is rapidly approaching us, with a current radial velocity of -119 km/sec -- nearly 7 times as fast as Voyager 1's (Sun-relative) motion.
According to the above-cited Wikipedia article, in 40,000 years, Gliese 445 will be about 3.45 light-years away (but still too dim to see with the naked eye), and Voyager 1 will be 1.6 light-years from it.