I've hired and fired people with these alphabet soup certs for about 10 yrs now, and I came up through the ranks myself. Apologies that this post is so long, but I want to tell you the straight scoop (especially beyond overated but necessary certs).
1. Certs only get you past the resume screening. Certs just suggest you have initiative, some professionalism, and can learn. What you have to know will basically be learned through OJT, because the real world has 1000s of permutations of systems and networks.
2. The best advice by far in this thread is build your own lab, except the lab you build for sysadm play is quite different from netadm play. A sysadm setup will have diverse networked and interoperable OS hosts (both DT & NB PCs), with numerous peripherals for learning backup, san, nas, etc. A netadm setup will be a minimal CCNP setup (you can get equip. list of cisco website, but buy used) plus a few extras. You'll need to set up a few hosts (to manage routing, etc.) and services (to manage b'cast impact, etc.), but not much more is needed from hosts.
3. Don't try to be expert at both sysadm & netadm at the same time; this can only come with time. Learn to do one well and learn just enough of the other to support the former. You are more valuable as an adm with "deep" knowledge than with "broad" knowledge.
4. People with excellent troubleshooting and problem-solving skills ALWAYS stand out and are highly valued.
5. If I need a sysadm, I don't care about cisco. If you really want to be a kick*ss sysadm beyond certs, learn adm basics (e.g., backups), service interoperability (winxp-linux-solaris-mac) and security hands-on (not from a book or in pristine environment) - - CANNOT GO WRONG WITH THIS.
6. If I need a netadm, I don't care too much (but just a little) about MS or Linux, except that he/she can configure/tune a host stack or instruct someone else. I need my netadm to know and apply fundamental design, performance and security principles - - I am so sick of CCNAs and CCNPs who only know a few cisco commands and how to connect boxes. Focus on hands-on design, troubleshooting, and security - - CANNOT GO WRONG WITH THIS.
7. If I need a combo of sys and net, I either (a) really need one skill set more than the other, (b) really need to do two hires but can't afford both, or (c) really want one hire to satisfy both but run the risk of a disastrous situation (over-worked, under-paid, my org too dependent on one person). What you want as a candidate is this: a chance to do one well, and a chance to learn the other.
8. As a candidate, you're better just presenting your real strengths and let them know you can fill-in-the-gaps because you've already mastered one-half of what's needed. I can't count the number of candidates who eliminated themselves because they talked as though they knew it all but ended up saying something stupid or absurd in the interview.