Let's see how much Data the Burners are writing:
1x ~ 1385kibibyte per second
16x ~ 22160kibibyte per second
3x 16x ~ 66480kibibyte per second
66480kibibyte * 1024 * 8 = 544604160bit per second
544604160bit / 10^6 ~ 545mbit per second
545mbit / 8 ~ 68mb per second
A drive that can sustain 68mb per second while seeking data for 3 burners is needed.
These numbers are based on the effective write speed and thus don't contain overhead (communication, buffering, error correction etc.) and compression.
Under the assumption that you have a IDE HDD running at full 100mb (purely theoretical), the DVD Burners could be saturated quite easily.
If you do burn a DVD at 16x and task your HDD with searching a file or some other work, it may not be able to feed enough data to a single Burner (unless it cached whatever you are burning into your RAM) - if it is an older HDD. Since you burn different files on each Burner the HDD is busy seeking and can't feed data with full speed. All under the assumption that you use a single HDD.
The picture changes a little by using a SATA 150mb HDD but probably not enough to supply 3 Burners.
Here´s a little chart i found on THG showing some bandwidth data.
http://www.tomshardware.com/2007/11/21/samsung_overtakes_with_a_bang/page9.html
As you can see, most drives can easily feed 68MB per second at max speed. What you need to look for is the minimum speed though. None of the drives can deliver what you need. Under perfect conditions the fastest of them might work but i really doubt that.
What you have to do now is to find where exactly the bottleneck is. It seems easy to blame a slow HDD but that may not be enough. If you connect all your burners to your mainboards PATA controler, you may be creating another bottleneck. The controller can only supply a limited set of information (the newest ATA norm makes it 133MB) which should be enough but doesn't have to be enough if it is using a different standard and is poorly implemented.
I think the fact that you burn 3 dvd with different content is creating this problem. The HDD has to seek different data for all three drives and thus can't fill it's bandwidth proberly since it's seeking all the time.
First you should test if it is really the HDD. Borrow an external drive or a large Flash memory module. Copy the files you want to burn on that medium and burn two DVDs at max speed using the HDD as a source for one burner and the other medium as a source for the second burner. If things go smoothly, a second hdd, RAID 0 or not, will do you a big favor.
If that's the case make a RAID 0. The seek times won't improve but two HDDs have an easier time filling the Burners buffers since they have a lot more bandwidth. That will increase throughput quite a bit. In addition make sure your HDD is cleaned up. Defragment it.
While the SATA bus can feed 150mb that does not mean your HDD can feed that much data. Even with a RAID 0 you might end up with your three burners running at 12x each.