13" MacBook Memory Issue

Hello, I'm new to this and wanted to see if someone else has had this issue and been able to resolve it.

Overview
I've had my MacBook for a few years. I recently replaced the Hard Drive from 80GB to 250GB without any issues. After replacing the Hard Drive, I got the idea to also replace the Memory. I read in Apple Support that I can upgrade from 1GB (2 slots of 512) to 4GB (2 slots of 2GB). Here's where I got that info>> MacBook 13" Technical Specs

My Problem
So I received my 4GB Memory (2x2GB) today. I turned off my MacBook, removed all connections, removed battery and removed and replaced the original memory. Each slot had 512 and replaced with the new 2GB. I put everything back together and turned on my MacBook. I heard it boot but the screen remained black. I shut it off by holding down the power button and tried again with no luck. I didn't hear anything except my RAM booting up.
I removed everything again to see if maybe the memory was loose but they weren't. I got an idea to replace one of the new 2GB with an old 512. To my surprise, it booted up and the screen turned on without an issue. I clicked on "About This Mac" to see if it read it correctly and it did.
(Please see below from "About This Mac").

Memory Slots:

ECC: Disabled

BANK 0/DIMM0:

Size: 512 MB
Type: DDR2 SDRAM
Speed: 667 MHz
Status: OK
Manufacturer: 0x2C00000000000000
Part Number: 0x38485446363436344844592D363637443320
Serial Number: 0xDA115BF0

BANK 1/DIMM1:

Size: 2 GB
Type: DDR2 SDRAM
Speed: 667 MHz
Status: OK
Manufacturer: 0x7FBA000000000000
Part Number: 0x000000000000000000000000000000000000
Serial Number: 0x00000000

It shows the 2GB plus the 512 for a total of 2.5GB.
So I got the idea that maybe one of the two new 2GB memory was bad.
I swapped the 2GB with the other one and it booted up without a problem as well. I then tried to replace the 512 with the 2GB and the same thing happened. (I heard it boot but the screen was black). I now think that it won't allow me to upgrade to 4GB. I'm confused because Apple Specs clearly states that I can upgrade to a max of 4GB.

Here's my Hardware Overview from "About This Mac".

Model Name: MacBook
Model Identifier: MacBook3,1
Processor Name: Intel Core 2 Duo
Processor Speed: 2 GHz
Number Of Processors: 1
Total Number Of Cores: 2
L2 Cache: 4 MB
Memory: 2.5 GB
Bus Speed: 800 MHz
Boot ROM Version: MB31.008E.B02
SMC Version (system): 1.24f3
Serial Number (system): W874676BZ62
Hardware UUID: 6538F157-F682-5CA6-8528-DE8C7DDCD6A3
Sudden Motion Sensor:
State: Enabled

Any help is greatly appreciated.

Thank you!

Ulises

Apple says your machine can take 2 x 2 GB modules for a total of 4 GB while MacTracker says you can actually install 6 GB.

Unless you got hold of some bad or mismatched memory it should have worked. Where did you purchase the memory from? If possible, send it back for a refund and buy your memory from Crucial or Other World Computing.