Hello Rocky:
1.) I modified the Sample Rate to be 100KHz (to make seing timings on the scope easier). I also tried Asynchronous mode. Also, The ADC clock is set to 48MHz (ADC_CR_CLKDIV(1))
LPC_SYSCON->ADCCLKSEL = 0;
LPC_SYSCON->ADCCLKDIV = 0;
Chip_ADC_Init(LPC_ADC, ADC_CR_RESOL(3) | ADC_CR_CLKDIV(1) | ADC_CR_ASYNC_MODE);
Note: Page 45 of rev. 1.4 of the user manual does not document the HALT or RESET bits. I discovered them in the debugger.
After doing this I get the same behavior. Going through 4 buffers always works. Linking the last descriptor back to the 1st breaks.
2.) I did actual think of widening the SCT trigger pulse. I tried a value that is 50% duty cycle and it had no effect.
3.) I set up some debug pins and have some scope plots:
I set the DMA Transfer size to 2. This made seing the signals a bit easier.
YELLOW : This gets toggled in the DMA transfer complete ISR
BLUE : This is the output of SCT0_7, the ADC start signal
PURPLE: I enabled the ADC conversion complete routine. The signal is toggled in the IRQ routine.
Plot 1:
I zoomed out so you could see the signals. You can see everything looks
YELLOW : This gets toggled in the DMA transfer complete ISR
BLUE : This is the output of SCT0_7, the ADC start signal
PURPLE: I enabled the ADC conversion complete routine. The signal is toggled in the IRQ routine.

Plot 2
I zoomed in to see the timings around the conversion start, the ADC complete and the DMA complete routines.
YELLOW : This gets toggled in the DMA transfer complete ISR
BLUE : This is the output of SCT0_7, the ADC start signal
PURPLE: I enabled the ADC conversion complete routine. The signal is toggled in the IRQ routine.

Plot 3
This is the Same as Plot 1 EXCEPT that the DMA transfer is only 1 ADC sample.
YELLOW : This gets toggled in the DMA transfer complete ISR
BLUE : This is the output of SCT0_7, the ADC start signal
PURPLE: I enabled the ADC conversion complete routine. The signal is toggled in the IRQ routine.

Plot 4:
This is Plot 3, just zoomed in around 1 set of pulses.
YELLOW : This gets toggled in the DMA transfer complete ISR
BLUE : This is the output of SCT0_7, the ADC start signal
PURPLE: I enabled the ADC conversion complete routine. The signal is toggled in the IRQ routine.

All of the plots make sense to me. The ADC triggers, it completes and then every 2 samples we get a DMA IRQ.
Let me know if there is anything else to try.... This very mysterious.
Thanks for the help.