There are several ways in which this can happen either by design or by strange happenings.
For AIM Logins (@mac.com @me.com names and AIM Names)
Go to iChat Menu > Preferences > Accounts > Security.
1) Setting Allow all should cure this if the setting is something else when you get there.
2) Setting Allow Buddy List only will only show those Buddies that you add to your Buddy List (meaning people trying your ID will not find you On Line)
2a) the reverse is true as well - if you Buddy sets this and removes you or does not keep you in their List then you can't "see" them either.
3) Same with Allow Specific Buddies Only
This can be useful using one Name for both home and work where only some Buddies are listed in this list (As well as running a Buddy List)
3a) same again if the other end has not got you in this list
4) They could have Blocked you (or you them) either in Block All or Specific Buddies.
You should check the list of Blocked people (And so should they)
5) PARENTAL CONTROLS
We are still talking about AIM logins
IF your Mac User Account can have Parental Controls turned On AND you set the Mail and iChat options then the moment you do this all Buddies are Blocked.
If you then set one Buddy as Allowed in the Parental Controls then they get added to the Allow Specific People List (and all others are effectivley Blocked).
5a) If you have an Admin Account but set up a test Mac User Account and enable Parental Controls then iChat will get blocked the same way As it is your Account/Buddy List at AIM that gets flagged.
When you "return" to using your Admin Account nay changes in the test/Parental Controls account will effect your Buddy List.
6) Strange and Incomplete Logins.
Sometimes when you login in to AIM your login can drop
This sometimes causes issues with the details in the relevant .plist and the corresponding info at the AIM servers.
How much this seems to interfere with your Login and retrieving info can depend on how the login issue arose.
As you say you have OS X 10.6.8 then go to yor Home Folder/Library/Preferences
Find the file called com.apple.ichat.AIM.plist and drag it to the Trash and then restart iChat.
(People with Lion and iChat 6 need to go to the Finder's Go Menu and hold the ALT key down to see the Library in the list as it is generally hidden).
7) Back to iChat itself.
One end could have set their Buddy List Status to Invisible and if this is done neither end can see the other (Like the lists about which Buddies can see you)
If it is not an AIM login of any sort please let us know what type it is.
(Yahoo in Lion/iChat 6 can also be Invisible)
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Lion 10.7.3)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
"Limit the Logs to the Bits above Binary Images." No, Seriously