i am having all sorts of weird things happening with this nic as well in Xubuntu 13.10. Sometimes websites cant even be resolved like github.com. It's very strange, it's autoloading the r8169 driver but my chipset is 
sudo lshw -C network
[sudo] password for ubu: 
  *-network               
       description: Ethernet interface
       product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller
       vendor: Realtek Semiconductor Co., Ltd.
       physical id: 0
       bus info: pci@0000:03:00.0
       logical name: eth2
       version: 0c
       serial: d4:3d:7e:e2:27:3b
       size: 100Mbit/s
       capacity: 1Gbit/s
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation
       configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168g-2_0.0.1 02/06/13 ip=192.168.0.13 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s
       resources: irq:43 ioport:d000(size=256) memory:f7c00000-f7c00fff memory:f0000000-f0003fff

03:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 0c)
	Subsystem: Micro-Star International Co., Ltd. Device 7850
	Flags: bus master, fast devsel, latency 0, IRQ 43
	I/O ports at d000 [size=256]
	Memory at f7c00000 (64-bit, non-prefetchable) [size=4K]
	Memory at f0000000 (64-bit, prefetchable) [size=16K]
	Capabilities: [40] Power Management version 3
	Capabilities: [50] MSI: Enable+ Count=1/1 Maskable- 64bit+
	Capabilities: [70] Express Endpoint, MSI 01
	Capabilities: [b0] MSI-X: Enable- Count=4 Masked-
	Capabilities: [d0] Vital Product Data
	Capabilities: [100] Advanced Error Reporting
	Capabilities: [140] Virtual Channel
	Capabilities: [160] Device Serial Number 01-00-00-00-68-4c-e0-00
	Capabilities: [170] Latency Tolerance Reporting
	Kernel driver in use: r8169

sudo ethtool eth2
Settings for eth2:
	Supported ports: [ TP MII ]
	Supported link modes:   10baseT/Half 10baseT/Full 
	                        100baseT/Half 100baseT/Full 
	                        1000baseT/Half 1000baseT/Full 
	Supported pause frame use: No
	Supports auto-negotiation: Yes
	Advertised link modes:  10baseT/Half 10baseT/Full 
	                        100baseT/Half 100baseT/Full 
	                        1000baseT/Full 
	Advertised pause frame use: Symmetric Receive-only
	Advertised auto-negotiation: Yes
	Link partner advertised link modes:  10baseT/Half 10baseT/Full 
	                                     100baseT/Half 100baseT/Full 
	Link partner advertised pause frame use: Symmetric Receive-only
	Link partner advertised auto-negotiation: Yes
	Speed: 100Mb/s
	Duplex: Full
	Port: MII
	PHYAD: 0
	Transceiver: internal
	Auto-negotiation: on
	Supports Wake-on: pumbg
	Wake-on: g
	Current message level: 0x00000033 (51)
			       drv probe ifdown ifup
	Link detected: yes


cat /var/log/syslog | grep r8169
Feb  1 21:15:32 saucy-haswell kernel: [    0.401021] r8169 Gigabit Ethernet driver 2.3LK-NAPI loaded
Feb  1 21:15:32 saucy-haswell kernel: [    0.401026] r8169 0000:03:00.0: can't disable ASPM; OS doesn't have ASPM control
Feb  1 21:15:32 saucy-haswell kernel: [    0.406345] r8169 0000:03:00.0: irq 43 for MSI/MSI-X
Feb  1 21:15:32 saucy-haswell kernel: [    0.406458] r8169 0000:03:00.0 eth0: RTL8168g/8111g at 0xffffc90000c38000, d4:3d:7e:e2:27:3b, XID 0c000800 IRQ 43
Feb  1 21:15:32 saucy-haswell kernel: [    0.406459] r8169 0000:03:00.0 eth0: jumbo features [frames: 9200 bytes, tx checksumming: ko]
Feb  1 21:15:32 saucy-haswell NetworkManager[718]: <info> (eth2): new Ethernet device (driver: 'r8169' ifindex: 2)
Feb  1 21:15:32 saucy-haswell kernel: [    1.863210] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:32 saucy-haswell kernel: [    1.863216] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:35 saucy-haswell kernel: [    4.502177] r8169 0000:03:00.0 eth2: link up
Feb  1 21:15:36 saucy-haswell kernel: [    5.503225] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:40 saucy-haswell kernel: [    8.987381] r8169 0000:03:00.0 eth2: link up
Feb  1 21:15:41 saucy-haswell kernel: [    9.988401] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:45 saucy-haswell kernel: [   14.389038] r8169 0000:03:00.0 eth2: link up
Feb  1 21:15:46 saucy-haswell kernel: [   15.374854] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:50 saucy-haswell kernel: [   18.879517] r8169 0000:03:00.0 eth2: link up
Feb  1 21:15:51 saucy-haswell kernel: [   19.881021] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:55 saucy-haswell kernel: [   24.398022] r8169 0000:03:00.0 eth2: link up
Feb  1 21:15:56 saucy-haswell kernel: [   25.391157] r8169 0000:03:00.0 eth2: link down
Feb  1 21:15:59 saucy-haswell kernel: [   27.986939] r8169 0000:03:00.0 eth2: link up
Feb  1 21:28:47 saucy-haswell kernel: [    0.402081] r8169 Gigabit Ethernet driver 2.3LK-NAPI loaded
Feb  1 21:28:47 saucy-haswell kernel: [    0.402086] r8169 0000:03:00.0: can't disable ASPM; OS doesn't have ASPM control
Feb  1 21:28:47 saucy-haswell kernel: [    0.406869] r8169 0000:03:00.0: irq 43 for MSI/MSI-X
Feb  1 21:28:47 saucy-haswell kernel: [    0.406980] r8169 0000:03:00.0 eth0: RTL8168g/8111g at 0xffffc90000c38000, d4:3d:7e:e2:27:3b, XID 0c000800 IRQ 43
Feb  1 21:28:47 saucy-haswell kernel: [    0.406982] r8169 0000:03:00.0 eth0: jumbo features [frames: 9200 bytes, tx checksumming: ko]
Feb  1 21:28:47 saucy-haswell NetworkManager[734]: <info> (eth2): new Ethernet device (driver: 'r8169' ifindex: 2)
Feb  1 21:28:47 saucy-haswell kernel: [    1.575879] r8169 0000:03:00.0 eth2: link down
Feb  1 21:28:47 saucy-haswell kernel: [    1.575927] r8169 0000:03:00.0 eth2: link down
Feb  1 21:28:50 saucy-haswell kernel: [    4.245928] r8169 0000:03:00.0 eth2: link up
Feb  1 21:28:51 saucy-haswell kernel: [    5.230723] r8169 0000:03:00.0 eth2: link down
Feb  1 21:28:54 saucy-haswell kernel: [    8.751562] r8169 0000:03:00.0 eth2: link up
Feb  1 21:28:55 saucy-haswell kernel: [    9.736904] r8169 0000:03:00.0 eth2: link down
Feb  1 21:29:00 saucy-haswell kernel: [   14.101403] r8169 0000:03:00.0 eth2: link up
Feb  1 21:29:01 saucy-haswell kernel: [   15.102398] r8169 0000:03:00.0 eth2: link down
Feb  1 21:29:04 saucy-haswell kernel: [   18.622758] r8169 0000:03:00.0 eth2: link up
Feb  1 21:29:05 saucy-haswell kernel: [   19.608568] r8169 0000:03:00.0 eth2: link down
Feb  1 21:29:10 saucy-haswell kernel: [   24.161134] r8169 0000:03:00.0 eth2: link up
Feb  1 21:29:11 saucy-haswell kernel: [   25.162696] r8169 0000:03:00.0 eth2: link down
Feb  1 21:29:13 saucy-haswell kernel: [   27.800400] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:05 saucy-haswell kernel: [ 1338.821698] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:08 saucy-haswell kernel: [ 1341.341979] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:09 saucy-haswell kernel: [ 1342.340383] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:12 saucy-haswell kernel: [ 1345.783690] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:13 saucy-haswell kernel: [ 1346.783681] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:18 saucy-haswell kernel: [ 1351.504926] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:19 saucy-haswell kernel: [ 1352.505480] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:22 saucy-haswell kernel: [ 1356.037329] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:23 saucy-haswell kernel: [ 1357.032605] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:28 saucy-haswell kernel: [ 1361.496623] r8169 0000:03:00.0 eth2: link up
Feb  1 21:51:29 saucy-haswell kernel: [ 1362.481940] r8169 0000:03:00.0 eth2: link down
Feb  1 21:51:32 saucy-haswell kernel: [ 1365.161561] r8169 0000:03:00.0 eth2: link up