Very good question. A simple answer would be that they can but with limitations.
There are numerous climate models, some quite simple and others that are incredibly complex, by and large there’s not really that much difference in accuracy. The main difference is that the complex models look at more aspects of the climate.
Indeed, the very first climate model was one developed in the 1970’s by the JASON’s and was simply called The Climate Model of the World. It’s predictions have been very accurate with all temperature predictions coming within 0.05°C of real-world values.
When it comes to actually recreating past temperatures, all models are very good at doing this. In producing the model, one of the things that the modellers will do is to produce hindcasts (opposite of forecasts). The models can be tweaked to accurately recreate the historical climate and in doing so, make them better at forecasting.
Even a basic model will factors in many variables including:
• The land↔atmosphere interaction
• The soil↔biosphere interaction
• The atmosphere↔biosphere interaction
• The ice↔ocean interaction
• The atmpsphere↔ice interaction
• Cryospherics: Glaciers, ice sheets, ice shelves, sea-ice, snow, permafrost, frozen ground etc
• Land surfaces: Vegetation, geomorphology, ecosystems, land use, land use change, albedo, volcanoes
• Oceanics: Biogeochemistry, thermohaline and other circulations, sea-levels
• Hydrology: Evapouration, transpiration, precipitation, cloud cover
• Weather: Wind stresses, wind dynamics, thermodynamics
• Oscillations in the oceans and atmosphere. Very long term models will take planetary oscillations into account.
• Radiation: Terrestrial and solar, changes to both
• Boundary transitions
• Changes in solar activity
• Changes in human activity
All the models will produce forecasts that include basics such as temperature and atmospheric changes, most models will also focus on some specific areas of the climate.
The overall accuracy varies greatly, some models have been way off the mark, others have been spot on. No model has been completely wrong, similarly none of them have been entirely accurate either. The early simple models perhaps had an accuracy of 70%, today’s complex models have about the same degree of accuracy. I wouldn’t like to single any one out as being the best or the worst as they all have their uses.
More info…
Climate Models: An Assessment of Strengths and Limitations
http://www.clivar.org/organization/wgcm/references/sap3-climate-models.pdf
Climate Models and Their Evaluation
http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-chapter8.pdf
The JASON’s climate model of the world (the first real climate model)
http://www.nap.edu/catalog.php?record_id=12181
IPCC Model Outputs
http://www.ipcc-data.org/ar4/gcm_data.html