Interestingly enough the Repository imagemagick package in Ubuntu 16.04 claims to support Adobe Digital Negative Raw (DNG) Image files:
andrew@corinth:~$ identify -list format | grep -i dng
3FR DNG r-- Hasselblad CFV/H3D39II
ARW DNG r-- Sony Alpha Raw Image Format
CR2 DNG r-- Canon Digital Camera Raw Image Format
CRW DNG r-- Canon Digital Camera Raw Image Format
DCR DNG r-- Kodak Digital Camera Raw Image File
DNG DNG r-- Digital Negative
ERF DNG r-- Epson RAW Format
K25 DNG r-- Kodak Digital Camera Raw Image Format
KDC DNG r-- Kodak Digital Camera Raw Image Format
MEF DNG r-- Mamiya Raw Image File
MRW DNG r-- Sony (Minolta) Raw Image File
NEF DNG r-- Nikon Digital SLR Camera Raw Image File
NRW DNG r-- Nikon Digital SLR Camera Raw Image File
ORF DNG r-- Olympus Digital Camera Raw Image File
PEF DNG r-- Pentax Electronic File
RAF DNG r-- Fuji CCD-RAW Graphic File
RAW DNG r-- Raw
RMF DNG r-- Raw Media Format
RW2 DNG r-- Panasonic Lumix Raw Image
SR2 DNG r-- Sony Raw Format 2
SRF DNG r-- Sony Raw Format
X3F DNG r-- Sigma Camera RAW Picture File
andrew@corinth:~$
However conversion of these files with imagemagick fails with a welter of error messages. This is confirmed on my own Ubuntu 16.04 installation as well as yours.
If you install the Unidentified Flying Raw (UFRaw) package though all will then be well:
sudo apt-get install ufraw-batch
This tested perfectly on my system, with my own sample file, and should also on yours (note especially the call to ufraw-batch in the Terminal output):
andrew@corinth:~/Desktop$ convert -verbose L1004220.DNG test.png
"ufraw-batch" --silent --create-id=also --out-type=png --out-depth=16 "--output=/tmp/magick-3253fgAn5Rixofwy.png" "/tmp/magick-3253IyazXOpb6yKj"
Found a generic profile, type exif
/tmp/magick-3253fgAn5Rixofwy.png PNG 5216x3472 5216x3472+0+0 16-bit sRGB 92.39MB 1.100u 0:01.110
L1004220.DNG=>/tmp/magick-3253fgAn5Rixofwy.png DNG 5216x3472 5216x3472+0+0 16-bit sRGB 92.39MB 0.010u 0:00.009
writing raw profile: type=exif, length=6064
L1004220.DNG=>test.png DNG 5216x3472 5216x3472+0+0 16-bit sRGB 92.19MB 6.090u 0:06.210
andrew@corinth:~/Desktop$