Making the Jump From Sysadmin to Network Administrator?

termdex asks: "I've been looking to move from systems administration to network administration for the last couple years but for some set of reasons networking seems to be an impregnable area of work. My experience has been like the often clichèd 'chicken and egg' scenario. Most employers aren't interested in candiates that lack serious network admin experience (ie: 80/20 network/other), but it would seem difficult to get that level of experience if you're currently a sysadmin. What advice can Slashdot readers offer as what works best in making lateral career moves? What experiences can you relate that shows difficulty or success?"