There's a tonne of articles online about fixing you graphics card by baking it in the oven. Supposedly this 'reflows' the solder to fix flaky connections. It's even been reported on sites such as Lifehacker. Does it work? Is it safe?
No, baking a broken graphics card will not necessarily fix it. It is not safe for the graphics card (may cause further problems), and may be a health risk.
It is possible to repair failed joints in a reflow oven. (Source)
Many options exist for repairing. In one embodiment the repairing can be repeating the reflow operation.
This article discusses repairing consumer electronics yourself. (They also disclose that they are a "company that specializes in the repairs of electronics", and that they offer great service and fair pricing - not the most reliable of sources)
As mentioned earlier, solder tends to melt at 183C/217C, and without proper equipment these temperatures in that environment will never be achieved. These fixes are highly advised against. While they may provide a temporary fix, the issue has been exacerbated by causing more damage to a perfectly working PCB component.
Problems with baking your broken graphics card in a standard oven at home:
The problem that you're trying to fix may not be a failed solder joint - it could be a software problem, or a failed hardware component.
Since you are not using a reflow oven with a set temperature profile, you may damage smaller components, as your oven temperature will not be accurate and consistent throughout.
This patent describes a "Reflow Minioven for Electrical Components" - a temperature controlled oven that follows a specific temperature ramp profile to minimise adverse effects such as thermal shock and PCB/component damage.
In reflow soldering typically a ramp-soak-spike heater panel recipe is used [snip]
The effect of hot spots:
[snip] forced convection can assist in reducing hot spots [snip]. As a result more uniform heating of the PCBs and the components is produced and more uniformly satisfactory results and better solder joints are achieved without damage due to overheating of small components. Source
There are plastic parts on graphics cards (Here's a patent for a fan-cooled card, which uses an injection moulded plastic shroud). Depending on what plastic is used and the temperature of your oven, these parts may melt/burn in the oven, which is likely to damage your graphics card, and may produce toxic fumes.
When plastic is subjected to fire, noxious and toxic fumes result. Source
If the graphics card (or a component on it) was released before RoHS was enforced (2003), the solder may be lead-based, and since lead is toxic, this is not a good idea to do in the same oven you cook food.
Exposure to solder flux fumes can be harmful:
[snip] evidence of a peripheral airways reaction, while working in the electronics industry and exposed to solder flux fumes containing colophony (pine resin). Source
If you do succeed in melting the solder, components may shift (causing short circuits, failed connections) or fall off.
Double sided products such as printed circuit boards are more difficult to reflow solder than single sided products since special care must be directed to the components already soldered on one side of the product to prevent them from shifting position or falling off while the other side is being reflow soldered. Source
In some cases this can repair it through causing solder to flow correcting any possible gaps. Occasionally a board can come from the manufacturer with defects such as cold solder joints (referenced to near the paragraph end) which can make a board prone to failure if it ever works at all.
There was a period where HP laptops suffered from a high rate of defects in their laptop video cards which would result in various symptoms. A search on motherboard baking or motherboard reflow will have a high count of HP models. Here they show a how to.
EDIT: To answer the "Is it safe?", I wouldn't do it with a card that is still functional as you can very well kill it. If it is done at low temperatures there is not likely to be any harm to the oven or baking person. The proper tool for something like this is actually more likely to be a heat gun however I am sure you are more likely to have an oven.
*I am in no way liable for any negative outcomes!
EDIT: I don't know why there would be a downvote since you are not going to find many sources on putting computer components in an oven. Anecdotal evidence is really all you are going to find.
A cold solder (or here) joint can be defined as "A solder connection exhibiting poor wetting and of greyish, porous appearance due to insufficient heat or excessive impurities in the solder. The condition can be caused by applying insufficient heat to the joint, inadequately cleaning the surfaces prior to soldering, insufficiently heating the part being soldered, improper tinning of the soldering iron tip, or poor heat control." which can actually occur in automated production. On a micro scale some characteristics described would be very difficult to visually see. This definition appears in IPC-T-50 (Interconnecting and Packaging Electronic Circuits - Terms and Definitions for Interconnecting and Packaging Electronic Circuits).
The HP nvidia graphics problem was wide spread enough to prompt a class action lawsuit (formerly at http://www.nvidiasettlement.com/ but referenced by http://fairnvidiasettlement.com/) which the complainants lost. They were seeking replacements with "similar value and kind" laptops however the judge ruled against them.