Update: I ran through the compatibility matrix after running these:
[root@localhost:~] lspci -v | grep -A1 -i ethernet
0000:01:00.0 Ethernet controller Network controller: Intel Corporation 82574L Gigabit Network Connection [vmnic0]
Class 0200: 8086:10d3
--
0000:02:00.0 Ethernet controller Network controller: Intel Corporation 82574L Gigabit Network Connection [vmnic1]
Class 0200: 8086:10d3
--
0000:05:00.0 Ethernet controller Network controller: Intel(R) Ethernet Controller X540-AT2 [vmnic2]
Class 0200: 8086:1528
--
0000:05:00.1 Ethernet controller Network controller: Intel(R) Ethernet Controller X540-AT2 [vmnic3]
Class 0200: 8086:1528
[root@localhost:~] lspci -n | grep 8086:1528
0000:05:00.0 Class 0200: 8086:1528 [vmnic2]
0000:05:00.1 Class 0200: 8086:1528 [vmnic3]
I ended up here:
https://my.vmware.com/group/vmware/details?downloadGroup=DT-ESXI55-INTEL-IXGBE-4.4.1&productId=353
I installed the driver on 5.5. Checking vmkernel.log I found this:
2016-05-05T05:23:50.921Z cpu13:33463)Loading module ixgbe ...
2016-05-05T05:23:50.924Z cpu13:33463)Elf: 1861: module ixgbe has license GPL
2016-05-05T05:23:50.927Z cpu13:33463)module heap: Initial heap size: 102400, max heap size: 68476928
2016-05-05T05:23:50.927Z cpu13:33463)vmklnx_module_mempool_init: Mempool max 68476928 being used for module: 4124
2016-05-05T05:23:50.927Z cpu13:33463)vmk_MemPoolCreate passed for 25 pages
2016-05-05T05:23:50.927Z cpu13:33463)skb_mem_info mempool for module ixgbe created - max size 134217728
2016-05-05T05:23:50.927Z cpu13:33463)module heap: using memType 0
2016-05-05T05:23:50.927Z cpu13:33463)module heap vmklnx_ixgbe: creation succeeded. id = 0x410a03ab1000
2016-05-05T05:23:50.927Z cpu13:33463)<6>Intel(R) 10 Gigabit PCI Express Network Driver - version 3.7.13.7.14iov-NAPI
2016-05-05T05:23:50.927Z cpu13:33463)<6>Copyright (c) 1999-2011 Intel Corporation.
2016-05-05T05:23:50.927Z cpu13:33463)PCI: driver ixgbe is looking for devices
2016-05-05T05:23:50.927Z cpu13:33463)DMA: 612: DMA Engine 'vmklnxpci-0:5:0.0' created using mapper 'DMANull'.
2016-05-05T05:23:50.927Z cpu13:33463)DMA: 612: DMA Engine 'vmklnxpci-0:5:0.0' created using mapper 'DMANull'.
2016-05-05T05:23:50.927Z cpu13:33463)DMA: 612: DMA Engine 'vmklnxpci-0:5:0.0' created using mapper 'DMANull'.
2016-05-05T05:23:50.927Z cpu13:33463)DMA: 657: DMA Engine 'vmklnxpci-0:5:0.0' destroyed.
2016-05-05T05:23:51.109Z cpu13:33463)<6>ixgbe: Enabled/Disable CNA set to 1
2016-05-05T05:23:51.109Z cpu13:33463)<6>ixgbe: 0000:05:00.0: ixgbe_check_options: CNA enabled, 1 queues
2016-05-05T05:23:51.109Z cpu13:33463)<6>ixgbe: 0000:05:00.0: ixgbe_check_options: Packet split is not supported.
2016-05-05T05:23:51.143Z cpu13:33463)<3>ixgbe 0000:05:00.0: The EEPROM Checksum Is Not Valid
2016-05-05T05:23:51.143Z cpu13:33463)WARNING: vmklinux: pci_announce_device:1488: PCI: driver ixgbe probe failed for device 0000:05:00.0
There you have it. Looks like a bad EEPROM/card.