This bug hit me after an upgrade from 14.04 to 16.04.1. Strangely I was not affected earlier. Maybe a reinstall might fix this issue? Two out of three reboots lead to failsafe X. Sleep command improved the situation but still 50%...

Other solutions like restart in rc.local somehow did not work. Maybe systemd is ignoring it or it is executed way to early. Sleep in /etc/init/lightdm.conf improved the situation but did not fix it.

$ lspci -nnvv
00:02.0 VGA compatible controller [0300]: Intel Corporation 4th Generation Core Processor Family Integrated Graphics Controller [8086:041e] (rev 06) (prog-if 00 [VGA controller])
        DeviceName:  Onboard IGD
        Subsystem: ASUSTeK Computer Inc. 4th Generation Core Processor Family Integrated Graphics Controller [1043:8534]
        Control: I/O+ Mem+ BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR- FastB2B- DisINTx+
        Status: Cap+ 66MHz- UDF- FastB2B+ ParErr- DEVSEL=fast >TAbort- <TAbort- <MAbort- >SERR- <PERR- INTx-
        Latency: 0
        Interrupt: pin A routed to IRQ 44
        Region 0: Memory at f7800000 (64-bit, non-prefetchable) [size=4M]
        Region 2: Memory at e0000000 (64-bit, prefetchable) [size=256M]
        Region 4: I/O ports at f000 [size=64]
        Expansion ROM at <unassigned> [disabled]
        Capabilities: [90] MSI: Enable+ Count=1/1 Maskable- 64bit-
                Address: fee0100c  Data: 41e1
        Capabilities: [d0] Power Management version 2
                Flags: PMEClk- DSI+ D1- D2- AuxCurrent=0mA PME(D0-,D1-,D2-,D3hot-,D3cold-)
                Status: D0 NoSoftRst- PME-Enable- DSel=0 DScale=0 PME-
        Capabilities: [a4] PCI Advanced Features
                AFCap: TP+ FLR+
                AFCtrl: FLR-
                AFStatus: TP-
        Kernel driver in use: i915
        Kernel modules: i915

CPU: Intel(R) Core(TM) i3-4160 CPU @ 3.60GHz

Symptoms:
/var/log/Xorg.0.log
[    67.571] (EE) intel(0): [drm] failed to set drm interface version: Permission denied [13].
[    67.571] (II) intel(0): [drm] Contents of '/sys/kernel/debug/dri/0/clients':
[    67.571] (II) intel(0): [drm] a dev pid    uid      magic     ioctls
[    67.571] (II) intel(0): [drm] 
[    67.571] (II) intel(0): [drm] y   0  2571     0          0       2004
[    67.571] (II) intel(0): [drm] y   0   179     0          0         20
[    67.571] (II) intel(0): [drm] y   0   179     0          0         19
[    67.571] (II) intel(0): [drm] y   0   179     0          0         19
[    67.571] (II) intel(0): [drm] y   0   179     0          0        116
[    67.571] (EE) intel(0): Failed to claim DRM device.

## xdiagnose process running after start
root      3073  0.0  0.0   7008  1560 ?        S    10:16   0:00 /bin/bash /usr/share/xdiagnose/failsafeXinit /etc/X11/xorg.conf.failsafe
##
root      2986  0.0  0.0   7108  1544 ?        Ss   10:16   0:00 /bin/bash /usr/share/xdiagnose/failsafeXServer
root      3001  0.0  0.0   3748   656 ?        S    10:16   0:00 xinit /usr/share/xdiagnose/failsafeXinit /etc/X11/xorg.conf.failsafe -- /usr/bin/X -br -once -config /etc/X11/xorg.conf.failsafe -logfile /var/log/Xorg.failsafe.log
root      3002  0.0  0.6 187192 25856 tty2     S<sl+ 10:16   0:00 /usr/lib/xorg/Xorg :0 -br -once -config /etc/X11/xorg.conf.failsafe -logfile /var/log/Xorg.failsafe.log
root      3073  0.0  0.0   7008  1560 ?        S    10:16   0:00 /bin/bash /usr/share/xdiagnose/failsafeXinit /etc/X11/xorg.conf.failsafe


gdm3 currently only comes up with a black screen and a mouse cursor and no error in the logs...
sddm is working fine but the lockscreen locks the terminal permanently, only killing the process on tty1 unlocked it.

Current workaround:

$ cat /etc/systemd/system/lightdmrestart.service
[Unit]
Description=Restart lightdm
After=graphical.target
Wants=lightdm.service accounts-daemon.service

[Service]
Type=oneshot
ExecStart=/usr/local/sbin/restartlightdm.sh

[Install]
WantedBy=default.target


$ cat/usr/local/sbin/restartlightdm.sh
#!/bin/bash

sleep 2
CHECK=$(ps axu | grep "/bin/bash /usr/share/xdiagnose/failsafe[X]init /etc/X11/xorg.conf.failsafe")
if [ ! -z "$CHECK" ]; then
  ps axu | grep "/bin/bash /usr/share/xdiagnose/failsafe[X]init /etc/X11/xorg.conf.failsafe" | awk '{print $2}' | xargs kill
  sleep 2
  systemctl restart lightdm
else
  exit 0
fi

systemctl enable lightdmrestart
systemctl reboot

Currently working without an issue. Hope it helps if someone else is still affected and needs a quick fix.