It's possible that network card used for your "second network" is no longer supported (i.e. could require purchasing and inserting new card from VMware's approved HCL) or perhaps you just need to update the driver, or configure the link/duplex, etc. Can you share the model of the NIC, server and switch? Perhaps someone has experience with that (even if it is a whitebox for example).
Some things you can review in the meantime are (from a console session on host):
~ # esxcfg-nics -lThe above will show you the driver, link duplex, etc. The 'esxcfg-nics' command can also set the speed duplex (just type --help for syntax) , but be careful not to kick yourself. That should be done from the console (i.e. standing in front of the server) or out via of band connection (i.e. iLO/DRAC/RSA).
~ # esxcli software vib list | lessThe above will show what ESXi software/VIB versions are installed (i.e. your NIC driver). Press PageDown to scroll through.
Or if you know what you are looking for you can grep for it. For example, the following will show me at a glance all the versions of my network card drivers:
~ # esxcli software vib list | grep -i 'tg\|qlc\|igb'
net-tg3 3.129d.v50.1-1OEM.500.0.0.472560 Broadcom VMwareCertified 2013-03-28
net-igb 4.0.17-1OEM.500.0.0.472560 Intel VMwareCertified 2013-03-28
net-qlcnic 5.0.750-1OEM.500.0.0.472560 QLogic VMwareCertified 2013-03-28
The objective is to compare your current driver version with the latest (or desired) version available from the VMware downloads page or from your hardware manufacturer's 3rd party VIB depot for example.
Also, If this is a home networking style switch, make sure the router has enough IP addresses in the DHCP scope and check for MAC security, etc. If it's a real switch and you are vlan tagging for example, that could be an easy one to miss if you forgot to configure your port group with the vlan tag. Also double check other obvious things like subnet mask, etc. Using the free RVTools reports are handy for this purpose (i.e. review the networking tabs such as "tabvNIC" and "tabvSC_VMK"). Compare the output of a working host vs non-working if you have that luxury. Couple ideas to get you started anyway. Good luck!