Somewhere in the sticky threads is a discussion which shows how to dump the contents of a serial EEPROM using a device programmer and an SOIC chip clip.

It seems to me that the programmer is powering everything on the PCB that connects to the same supply rail. I wonder why that doesn't trigger the programmer's overcurrent detection/protection logic.

Secondly, how can one be certain that the MCU is not competing with the programmer for access to the EEPROM? In my ADSL modem, for example, when its serial EEPROM is being externally programmed, the PCB's reset line is held low, thus tristating the CPU.

Does the MCU read the EEPROM on power-up, transfer the contents to RAM, and then ignore it until the next power cycle?

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A backup a day keeps DR away.

Depends on the EEPROM you are talking about as well as the programmer used, AND ALSO, the PCB. Personally, I use an adapter that has basic capacitors, resistors, etc. to allow writing to serial EEPROM, regardless of interference from nearby circuits. Worst cases, you can cut one leg of EEPROM only, then solder back when done. FAST. EASY.

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http://pcrecoveryllc.com

...or desolder with SMD equipment, read in programmer through SO/SOIC adapter, resolder. Safer and professional. We have all the time in the world (now) ...

Sometime you don't even need a programmer!

For a 93C* series eeprom you can connect the chip directly to the parallel port.
It will usually work with the ic in circuit. Just solder the leads to each pin.
Something like this:
1 (CS) pin 3 (D1)
2 (CLK) pin 5 (D0)
3 (DI) pin 6 (D2)
4 (D0) pin 12 (ACK)
5 (Vss) pin 23 (GND)
6 ( " ) pin 23 (GND)
7 (N/C) none
8 (Vcc) pin 2 (D3)

It's a "Wile E. Coyote" solution : if you don't have grounded and fine tip soldering iron and adequate wires you will most likely damage the pcb, especially on some materials used on some drives... *cough! cough!*

Yes a grounded temperature controlled Weller iron with a ptr8 tip.
A magnifying lamp.

If I have any concerns I sometimes remove the chip with hot air and then solder on the leads for reading/programming. And then I replace the chip.
I find it easier and quicker than using the Willem and i've never had any problems using this method.

And yes you would need top quality soldering skills to achieve good results and not cause any damage. I am very confident of my own skills in this department.

I am quite always right : the problem is that people don't understand, and they think it is simple even with a quick soldering gun or "Joe's Hardware store " china quality 5$ soldering iron...

Sometimes , something goes wrong even with my professional tools (PCB tend to delaminate especially if new Pb Free alloy is used) or a track needs to be repaired later... we call it "SFIGA" (a stroke of bad luck , or practical demonstration of Murphy's law).