The 380 has the Intel G41 Express chipset which supports the following CPUs:

http://www.cpu-upgrade.com/mb-Intel_(chipsets)/G41_Express.html

The graphics card would depend on which chassis type you have, you can find more information here:

http://www.dell.com/support/home/us/en/04/product-support/product/optiplex-380/manuals

http://www.dell.com/downloads/global/products/optix/en/optiplex-380-tech-guide.pdf

Intel® G41Express Chipset w/ICH7

Page 10 Tech Guide

Intel® Core™ 2 Quad Q9650/3.00GHz, 12M, 1333FSB
Intel® Core™ 2 Quad Q9550/2.83GHz, 12M, 1333FSB
Intel® Core™ 2 Quad Q9505/2.83GHz,  6M , 1333FSB
Intel® Core™ 2 Quad Q9400/2.66GHz,  6M, 1333FSB
Intel® Core™ 2 Quad Q8400/2.66GHz,  4M, 1333FSB

Intel Core 2 Duo
E4300 - HH80557PG0332M / BX80557E4300

E4300, 1.8 GHz, FSB 800 MHz, L2 2 MB
E4400 - HH80557PG0412M / BX80557E4400

E4400, 2 GHz, FSB 800 MHz, L2 2 MB
E4500 - HH80557PG0492M / BX80557E4500

E4500, 2.2 GHz, FSB 800 MHz, L2 2 MB
E4600 - HH80557PG0562M / BX80557E4600

E4600, 2.4 GHz, FSB 800 MHz, L2 2 MB
E4700 - HH80557PG0642M / BX80557E4700

E4700, 2.6 GHz, FSB 800 MHz, L2 2 MB
E6300 - HH80557PH0362M / BX80557E6300 / BX80557E6300T2

E6300, 1.86 GHz, FSB 1066 MHz, L2 2 MB
E6305
1.86 GHz, FSB 1066 MHz, L2 2 MB
E6320 - HH80557PH0364M / BX80557E6320

E6320, 1.86 GHz, FSB 1066 MHz, L2 4 MB
E6400 - HH80557PH0462M / BX80557E6400

E6400, 2.13 GHz, FSB 1066 MHz, L2 2 MB
E6405
2.13 GHz, FSB 1066 MHz, L2 2 MB
E6420 - HH80557PH0464M / BX80557E6420

E6420, 2.13 GHz, FSB 1066 MHz, L2 4 MB
E6540 - HH80557PJ0534M

E6540, 2.33 GHz, FSB 1333 MHz, L2 4 MB
E6550 - HH80557PJ0534MG / BX80557E6550 / BX80557E6550R

E6550, 2.33 GHz, FSB 1333 MHz, L2 4 MB
E6600 - HH80557PH0564M / BX80557E6600

E6600, 2.4 GHz, FSB 1066 MHz, L2 4 MB
E6700 - HH80557PH0674M / BX80557E6700

E6700, 2.66 GHz, FSB 1066 MHz, L2 4 MB
E6750 - HH80557PJ0674MG / BX80557E6750

E6750, 2.66 GHz, FSB 1333 MHz, L2 4 MB
E6850 - HH80557PJ0804MG / BX80557E6850

E6850, 3 GHz, FSB 1333 MHz, L2 4 MB
E7200 - AT80571PH0613M / EU80571PH0613M / BX80571E7200 / BXC80571E7200

E7200, 2.53 GHz, FSB 1066 MHz, L2 3 MB
E7300 - AT80571PH0673M / EU80571PH0673M / BX80571E7300 / BXC80571E7300

E7300, 2.66 GHz, FSB 1066 MHz, L2 3 MB
E7400 - AT80571PH0723M / AT80571PH0723ML / BX80571E7400 / BXC80571E7400

E7400, 2.8 GHz, FSB 1066 MHz, L2 3 MB
E7500 - AT80571PH0773M / AT80571PH0773ML / BX80571E7500 / BXC80571E7500

E7500, 2.93 GHz, FSB 1066 MHz, L2 3 MB
E7600 - AT80571PH0833ML / BX80571E7600 / BXC80571E7600

E7600, 3.06 GHz, FSB 1066 MHz, L2 3 MB
E8190 - EU80570PJ0676MN / BX80570E8190

E8190, 2.66 GHz, FSB 1333 MHz, L2 6 MB
E8200 - EU80570AJ0676M / EU80570PJ0676M / BX80570E8200 / BX80570E8200A / BXC80570E8200

E8200, 2.66 GHz, FSB 1333 MHz, L2 6 MB
E8300 - EU80570AJ0736M / EU80570PJ0736M

E8300, 2.83 GHz, FSB 1333 MHz, L2 6 MB
E8400 - AT80570PJ0806M / EU80570PJ0806M / BX80570E8400 / BX80570E8400A / BXC80570E8400

E8400, 3 GHz, FSB 1333 MHz, L2 6 MB
E8500 - AT80570PJ0876M / EU80570PJ0876M / BX80570E8500 / BX80570E8500A / BXC80570E8500

E8500, 3.16 GHz, FSB 1333 MHz, L2 6 MB
E8600 - AT80570PJ0936M / BX80570E8600 / BXC80570E8600

E8600, 3.33 GHz, FSB 1333 MHz, L2 6 MB

Intel Core 2 Extreme
QX9650 - EU80569XJ080NL / BX80569QX9650

QX9650, 3 GHz, FSB 1333 MHz, L2 12 MB
Intel Core 2 Quad
Q6600 - HH80562PH0568M / BX80562Q6600

Q6600, 2.4 GHz, FSB 1066 MHz, L2 8 MB
Q6700 - HH80562PH0678MK / BX80562Q6700

Q6700, 2.66 GHz, FSB 1066 MHz, L2 8 MB
Q8200 - AT80580PJ0534MN / AT80580PJ053HN / EU80580PJ0534M / EU80580PJ0534MN / BX80580Q8200 / BXC80580Q8200

Q8200, 2.33 GHz, FSB 1333 MHz, L2 4 MB
Q8200S - AT80580AJ0534MN / BX80580Q8200S

Q8200S, 2.33 GHz, FSB 1333 MHz, L2 4 MB
Q8300 - AT80580PJ0604ML / AT80580PJ0604MN / BX80580Q8300 / BXC80580Q8300

Q8300, 2.5 GHz, FSB 1333 MHz, L2 4 MB
Q8400 - AT80580PJ0674ML / AT80580PJ067HL / BX80580Q8400 / BXC80580Q8400

Q8400, 2.66 GHz, FSB 1333 MHz, L2 4 MB
Q8400S - AT80580AJ0674ML / AT80580AJ067HL / BX80580Q8400S

Q8400S, 2.66 GHz, FSB 1333 MHz, L2 4 MB
Q9300 - EU80580PJ0606M / BX80580Q9300

Q9300, 2.5 GHz, FSB 1333 MHz, L2 6 MB
Q9400 - AT80580PJ0676M / BX80580Q9400 / BXC80580Q9400

Q9400, 2.66 GHz, FSB 1333 MHz, L2 6 MB
Q9400S - AT80580AJ0676M / BX80580Q9400S

Q9400S, 2.66 GHz, FSB 1333 MHz, L2 6 MB
Q9450 - EU80569PJ067N / BX80569Q9450 / BXC80569Q9450

Q9450, 2.66 GHz, FSB 1333 MHz, L2 12 MB
Q9550 - AT80569PJ073N / EU80569PJ073N / BX80569Q9550 / BXC80569Q9550

Q9550, 2.83 GHz, FSB 1333 MHz, L2 12 MB
Q9550S - AT80569AJ073N / BX80569Q9550S

Q9550S, 2.83 GHz, FSB 1333 MHz, L2 12 MB
Q9650 - AT80569PJ080N / BX80569Q9650 / BXC80569Q9650

Q9650, 3 GHz, FSB 1333 MHz, L2 12 MB
Intel Pentium Dual-Core
E2140 - HH80557PG0251M / BX80557E2140 / BXC80557E2140

E2140, 1.6 GHz, FSB 800 MHz, L2 1 MB
E2160 - HH80557PG0331M / BX80557E2160 / BXC80557E2160

E2160, 1.8 GHz, FSB 800 MHz, L2 1 MB
E2180 - HH80557PG0411M / BX80557E2180 / BXC80557E2180

E2180, 2 GHz, FSB 800 MHz, L2 1 MB
E2200 - HH80557PG0491M / BX80557E2200 / BXC80557E2200

E2200, 2.2 GHz, FSB 800 MHz, L2 1 MB
E2210 - AT80571RG0491M / EU80571RG0491M / BXC80571E2210

E2210, 2.2 GHz, FSB 800 MHz, L2 1 MB
E2220 - HH80557PG0561M / BX80557E2220 / BXC80557E2220

E2220, 2.4 GHz, FSB 800 MHz, L2 1 MB
E5200 - AT80571PG0602M / EU80571PG0602M / BX80571E5200 / BXC80571E5200

E5200, 2.5 GHz, FSB 800 MHz, L2 2 MB
E5300 - AT80571PG0642M / AT80571PG0642ML / BX80571E5300 / BXC80571E5300

E5300, 2.6 GHz, FSB 800 MHz, L2 2 MB
E5400 - AT80571PG0682M / AT80571PG0682ML / BX80571E5400 / BXC80571E5400

E5400, 2.7 GHz, FSB 800 MHz, L2 2 MB
E5500 - AT80571PG0722ML / BX80571E5500 / BXC80571E5500

E5500, 2.8 GHz, FSB 800 MHz, L2 2 MB
E5700 - AT80571PG0802ML / BX80571E5700 / BXC80571E5700

E5700, 3 GHz, FSB 800 MHz, L2 2 MB
E5800 - AT80571PG0882ML / BX80571E5800

E5800, 3.2 GHz, FSB 800 MHz, L2 2 MB
E6300 - AT80571PH0722ML / BX80571E6300 / BXC80571E6300

E6300, 2.8 GHz, FSB 1066 MHz, L2 2 MB
E6500 - AT80571PH0772ML / BX80571E6500 / BXC80571E6500

E6500, 2.93 GHz, FSB 1066 MHz, L2 2 MB
E6500K - AT80571XH0772ML / BXC80571E6500K

E6500K, 2.93 GHz, FSB 1066 MHz, L2 2 MB
E6600 - AT80571PH0832ML / BX80571E6600 / BXC80571E6600

E6600, 3.06 GHz, FSB 1066 MHz, L2 2 MB
E6700 - AT80571PH0882ML / BX80571E6700 / BXC80571E6700

E6700, 3.2 GHz, FSB 1066 MHz, L2 2 MB
E6800 - AT80571PH0932ML / BX80571E6800 / BXC80571E6800

E6800, 3.33 GHz, FSB 1066 MHz, L2 2 MB

Yeah, but what the book says and what actually works is (apparently) two different things (due to FSC 1066/1333 issues) ... unless ... could it be the pre-installed ram throwing things off?

https://www.dell.com/community/Optiplex/FSB-of-Optiplex-380-MB/m-p/6047900#M23808

ANYWAY, what model Intel processor is currently installed?

 I had success with an X5470 Xeon conversion. This requires a notched Xeon CPU, an LGA771to LGA775 CPU tape adapter, and a modded BIOS from BIOSMods.com. Of course this is not a factory approved modification.

You will need to  install one of the heatpipe coolers. T9303 with a shroud from an optiplex 780 should work.

Delidded.com has instructions.

Maximum RAM is 8GB DDR3 @1066 speed. It must be x64 low density RAM. Crucial .com has these.

The ZOTAC GTX1050 Min 2GB fits and runs. The GTX1050Ti Mini 4GB version works also.

The QX9650 didn't work for me.

QX9650 works fine in Precision T3400 with current bios and 525W power supply.

https://www.dell.com/downloads/global/products/precn/en/precn_t3400_specsheet.pdf