I'm also seeing the same problem on a Dell XPS 13 9343 - Wifi is suddenly non-functional, whereas it'd otherwise been working fine for many years (I'm unsure exactly when the problem started, as I mostly use my laptop via wired ethernet, but it's definitely broken within the last couple of weeks).

I've tried booting the previous kernel 3.19.0-80-generic from the grub boot menu, but that also results in non-functional wifi. I've included (hopefully) relevant info below, but please let me know if there's anything else I can do to help.

$ cat /etc/issue
Ubuntu 14.04.5 LTS \n \l

$ uname -a
Linux shyknee 3.19.0-82-generic #90~14.04.1-Ubuntu SMP Thu Feb 23 01:12:44 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux

$ sudo lshw -C network
  *-network UNCLAIMED     
       description: Network controller
       product: BCM4352 802.11ac Wireless Network Adapter
       vendor: Broadcom Corporation
       physical id: 0
       bus info: pci@0000:02:00.0
       version: 03
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress cap_list
       configuration: latency=0
       resources: memory:f7200000-f7207fff memory:f7000000-f71fffff
  *-network
       description: Ethernet interface
       physical id: 2
       logical name: eth1
       serial: 00:50:b6:0f:5d:ca
       size: 100Mbit/s
       capacity: 100Mbit/s
       capabilities: ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation
       configuration: autonegotiation=on broadcast=yes driver=asix driverversion=22-Dec-2011 duplex=full firmware=ASIX AX88772B USB 2.0 Ethernet ip=192.168.0.54 link=yes multicast=yes port=MII speed=100Mbit/s

$ sudo apt-get purge bcmwl-kernel-source
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following packages will be REMOVED
  bcmwl-kernel-source*
0 to upgrade, 0 to newly install, 1 to remove and 0 not to upgrade.
After this operation, 8,045 kB disk space will be freed.
Do you want to continue? [Y/n] 
(Reading database ... 483437 files and directories currently installed.)
Removing bcmwl-kernel-source (6.30.223.248+bdcom-0ubuntu0.2) ...
Removing all DKMS Modules
Done.
update-initramfs: deferring update (trigger activated)
Purging configuration files for bcmwl-kernel-source (6.30.223.248+bdcom-0ubuntu0.2) ...
update-initramfs: deferring update (trigger activated)
Processing triggers for initramfs-tools (0.103ubuntu4.11) ...
update-initramfs: Generating /boot/initrd.img-3.19.0-82-generic

$ sudo apt-get install bcmwl-kernel-source
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following NEW packages will be installed
  bcmwl-kernel-source
0 to upgrade, 1 to newly install, 0 to remove and 0 not to upgrade.
Need to get 0 B/1,511 kB of archives.
After this operation, 8,045 kB of additional disk space will be used.
Selecting previously unselected package bcmwl-kernel-source.
(Reading database ... 483359 files and directories currently installed.)
Preparing to unpack .../bcmwl-kernel-source_6.30.223.248+bdcom-0ubuntu0.2_amd64.deb ...
Unpacking bcmwl-kernel-source (6.30.223.248+bdcom-0ubuntu0.2) ...
Setting up bcmwl-kernel-source (6.30.223.248+bdcom-0ubuntu0.2) ...
Loading new bcmwl-6.30.223.248+bdcom DKMS files...
First Installation: checking all kernels...
Building only for 3.19.0-82-generic
Building for architecture x86_64
Building initial module for 3.19.0-82-generic
Secure Boot not enabled on this system.
Done.

wl:
Running module version sanity check.
 - Original module
   - No original module exists within this kernel
 - Installation
   - Installing to /lib/modules/3.19.0-82-generic/updates/dkms/

depmod....

DKMS: install completed.
modprobe: ERROR: could not insert 'wl': Package not installed
update-initramfs: deferring update (trigger activated)
Processing triggers for initramfs-tools (0.103ubuntu4.11) ...
update-initramfs: Generating /boot/initrd.img-3.19.0-82-generic