I installed OpenLDAP 2.3.43-3.el5 on a CentOS 5.3 box. I configured slapd.conf:

suffix "dc=my-domain,dc=com"
rootdn "cn=Manager,dc=my-domain,dc=com"

I then grabbed LDAPAdmin which I've used before. I gave LDAPAdmin the rootdn as username, the password that I slappasswd'ed, and the "suffix" as "Base". When I hit "Test connection", it's successful. But when I connect, I immediately get a message saying, "LDAP Error: No such object!" Same error if I try to add a user. I'm not sure what "object" doesn't exist, and how to either summon it into existence or get LDAPAdmin to stop worrying about it.

ETA: I right-clicked on the LDAP server and chose Properties, and got:

Google isn't finding that. It translates as, "Access injury with address 0044F10C in module “LdapAdmin.exe”. Read from address 00000268."

You should be able to look at the server logs to give you more details.
For OpenLDAP, it's usually in /var/log/messages

Also, I'm sure you just obscured your real domain from us by putting my-domain in the examples above, but you have your real domain in the actual config files, right?

There is nothing in /var/log/messages Nor any other logs in /var/log/

Yes