I'm hosting a 16 player LAN party in July.

What games do you suggest?

Currently on the roster:

Chivalry
Carmageddon Max (with 32 player mod)
Renegade X
Rune
Blur

4-8 players would make it much easier to recommend games. Anyway...

Unreal Tournament '99
Red Faction

Jedi Academy (up to 32 players per server)

Yup. Get multiple PCs and do a roundrobin or elimination tournament with 4-8 players per machine.

Unreal Tournament, Quake 3 arena.

I liked Counterstrike.

Din's Curse has no cap on the number of people one can play with (hardware and connection will determine the actual upper limit). I think a lot of Soldak's other games likewise have either no limits, or high limits (16-32 players). They're on sale right now, too. :) (Just pick one -- they're all basically variations on the exact same gameplay.)

I think Terraria might allow 16 or more, but I don't think it has a proper LAN play option.

Mechwarrior 2 NETMECH multiplayer was released for free back in 1996, and allows up to 32 players on a map,
and more server/gameplay options than Mechwarrior Online.

All you need to do is visit mech2.org to downlaod the full version of NETMECH (complete with ultra-custom mech loadouts), and/or setup a IPX server on one of the machines in your LAN party.

Doing the IPX server thing is very easy, IPX emulation is embedded in DOSBox, and just requires a standalone DOSBox session to be open

This.

Call of Duty (for vehicles), Call of Duty 2 (more slow) or Call of Duty 4 (rather arcade-y). Unreal Tournament 2004 (Onslaught mode) is also really fun.

Serious Sam (The original) is quite a party, due to the way the respawns work.