I have been inactive on FFXIV because i had these errors 6 months ago, also after such short intervals each times, like my fellow gamers in this thread. I thought i'd give it another shot, hoping it would be resolved after such many patches or updates (because i reported it back then too, but no reply or answer came, let alone a solution to it)... But it's still the same issue...
I just hope that you are actually going to make good work/use of this case this time... Because the game is anything but fun, when you get disconnected every 5-15 minutes. My PC is a monster, my internet is one of the fastest available,... So it's not an issue of "your PC can't handle it". It runs all of the HD/HQ games just fine.

I'm a FFF (Final Fantasy Fan) and i think it would be a shame that the game would lose so many fans just because this issue can't be resolved or isn't focussed on. (Been "working/resolving it" for many months now and still no solution...)

So here's my data:
1) User Environment
1. – Operating System?
Windows Vista Home Premium 64 bit

2. - Antivirus program? (IE: Windows Defender, AVG, etc)
Kaspersky Internet Security 2010

3. – Internet Service Provider?
Telenet

2) Exact Error Message
Case 1: An error has occurred on the game server. (30002)1319722060,13345497,5
Case 2: An error has occurred on the game server. (30002)1319723910,13345497,5

3) Frequency of 30002 Error?
Sometimes after 5-15 minutes, one rare occasion of about 1 hour.

4) What area were you in when the error occurred?
La Noscea (Case 1 & 2)

5) What were you doing when the error occurred? (What type of actions, movements, etc)
Case 1: Talking to NPC (turning in Leve)
Case 2: Just finished editing macro's, not doing anything at that moment really...

Just like my fellow gamers mentioned, the action doesn't matter appearently. Wether it's fighting or exploring busy areas or being idle for the whole period.

6) Were you in a party at the time the error occurred? (Please provide number of party members if you were in a party)
No (Case 1 & 2)

7) Estimate number of characters that were near you at time of error?
Case 1 & 2: 1-3 people at most

8) Were you able to log back into FFXIV after the error?
When it kicks you back to Character Selection screen, when i then select the same character again it gives a Lobby error and kicks me back another screen: the Title Screen. When i then select to go to the character screen again and select my character, it logs back in again.
Also, letting it wait on the character screen for a few minutes in the first place is also good enough to let me log back in.
So to sum up: Yes, but we need to wait a few minutes to log back in.