Presently I am running bitcoin-qt 0.5.1 on ubuntu 11.04, and am running with seventeen peers. But if I run bitcoind (still 0.5.1 on ubuntu 11.04) I can't get more than eight, no matter how long I leave it on. Why is that, and should I do anything about it?
8 peers is the number of outbound connections the client makes. So if you always have 8 connections, that probably means that people can't make inbound connections to your machine. This really has no harmful effects. You can't at all trust people who connected to you anyway. (You might have 70 inbound connections that are all from the same miscreant.)
The most likely reason for the difference is that one client has UPNP support and the other doesn't or doesn't have it enabled. If your machine is behind a NAT device that supports UPNP, that's probably the reason.
If you want to enable inbound connections and thus have more than 8, most likely what you need to do is forward port 8333 on your router.
The number of connections does not really matter, but you should still have more than one (else that one could be a malicious person).
Also we could not imagine everybody connected to everybody in the network, this is why a small number is more conceivable if your not a miner you just need to download new blocks and push transactions which will be transfer across the network by the node you are connected and so on.