Replaced power supply, didn't help.

You might check here ... http://www.techspot.com/vb/topic95391.html
6 beeps does not look good... points to issues on the main board.
Maybe a replaceable chip, maybe not.

Dimension 4600 beeps, won't boot

Thank you.
I checked the sequence of the beeps, and it is: 2 long, 2 short, 1 long, 1 short.
I don't see that on the list?? And the diagnostic lights on the back : A & C are green, B & D are orange(?)
Any suggestions??

Your light code announces a memory issue, per
http://support.dell.com/support/edocs/systems/dim4600/en/4600/sm/advtrbl.htm#1084976

You might try reseating your memory, swapping the memory between available slots,
and trying the modules one at a time, to see if you can isolate the problem.
Memtest is also a good tool, but at this point I think it will only confirm what your light code has announced.

The beep code as you describe it does not appear on the page for your system, shown above.

If you need your Owners Manual, see this Link http://support.dell.com/support/edocs/systems/dim4600/en/4600i/index.htm

In other Dell Systems, your beep code means "Incompatible SDRAM DIMMs are installed."
You need memory that is compatible with your Dell, for example - http://www.kahlon.com/rm36497_Dell_Dimension_4600.html

In another thread (nearly two years old) raybay helped someone else with your system (4600) and noted how solid and dependable it is.

Dimension 4600 beeps, won't boot

Can you tell me how to reseat the memory, or swap between slots??

And regarding the beeps, we haven't changed anything, how coud it be "Incompatible SDRAM DIMMs are installed"?? And is this something I could do myself?

Thanks for your time, and wisdom.

Don't see a six beep alarm on the Dell site...

This means that this note about beep codes may apply in some Dells, but may not apply in all...
the 2-2-1-1 reference that I found was for a Dimension™ 900 which is NOT your system!
http://support.dell.com/support/edocs/systems/dcaydim2/codemess.htm#post_beep_codes

Again, though, it points to memory. Memory, like all hardware, can fail.

If you do not have one already, get your owner's manual here,
http://support.dell.com/support/edocs/systems/dim4600/en/series_en.htm

There are two different versions of your computer, one with integrated graphics, and the other with an installed graphics card.
The manuals differ and the page numbers differ.

This is something you can learn to do... although
if you are uncomfortable doing it... your local tech shop would be glad to have your business.
They might even provide some education in the process.

Read carefully, and understand the dangers of Static Electricity and How to avoid damaging your computer.
Note also how the memory modules are secured.
You will need to Carefully release them, and then reinstall them.
Note which slot is primary, and install only one at a time, initially, to see if the problem can be isolated to one module.