The quick answer is no.
BART requires separate tracks and infrastructure (because it runs on non-standard gauge tracks). To replace Caltrain with BART, all the Caltrain infrastructure would have to be removed and rail service would have to be shut down rail service for years during the construction period. Replacing the entire Caltrain line with BART could cost as much as twenty billion dollars and would take from thirty to fifty years to fund and construct. So it's out of the question.
Implicit in this question, however, is an assumption that it would be desirable to replace Caltrain with BART, that BART has positive qualities that Caltrain cannot hope to attain. We believe this assumption is based on erroneous impressions.
First, people like that BART runs frequently and somehow think that Caltrain cannot run frequently. Service frequency has more to do with providing the necessary funds to operate the system than technology per se. However, electrification is known to reduce operating costs and thereby allow transit agencies to run more trains (and frequent service) at less cost. Caltrain could run just as frequently, or more frequently than BART if funds were provided to operate the extra trains and if it is electrified to allow the trains to run more quickly.
Second, some people think that Caltrain is slower than BART. This perception was more common before Caltrain began running its "Baby Bullet" express service. In actuality, the average speeds of the two systems are about the same, about 34 mph (including stops), with BART slightly faster. Electrification and signal system upgrades will make Caltrain run even faster.
Third, some people just think that it would be better to have one major rail system for the Bay Area, that this would somehow make it easier for people to take transit. Such people are obviously not thinking about the extra travel time that would be introduced for many riders with the all-stops BART system, that would make taking transit less appealing and convenient compared to driving. They are also not realizing that many BART riders have to make transfers to other BART lines already, and that numbers of transfers would not necessarily be reduced. To have a single system for the Bay Area would also incapacitate more of the Bay Area the next time BART workers go on strike.
BART is a one-off system that requires custom-made trains that are incompatible with the worldwide standard that Caltrain electrification will follow. This decreases competition to build BART trains and increases BART’s cost. Caltrain, on the other hand, can operate off-the-shelf equipment produced by a variety of manufacturers worldwide. In fact, some of the nicest amenity-filled high-speed trains in Europe cost less to build than a BART train does.
BART has limited capacity in its cars due to its single level design, thus resulting in sardine-can like trains during peak periods. Also, BART lacks the ability to run express trains like Caltrain's "Baby Bullet" service.
Because it's powered by a third rail in the ground that will electrocute anyone who walks on it, BART requires the entire line to be grade-separated before a single train can run. Caltrain evolved with the historical conditions on the Peninsula and has a number of at-grade crossings. Converting these to grade-separated crossings would indeed be costly, usually in the range of $100 - $300 million per crossing, as it would have to be constructed in a manner that existing service is not disrupted. Grade separations are often controversial within communities, because some people do not want the noise and disruption created by construction, and some object to the aesthetics of building a "great wall" that visually divides the community. Others feel that it is an important safety and traffic issue and worth the cost. Regardless, the point is that with an electrified Caltrain each community can decide for itself whether or not to build a grade separation, and as money becomes available without stopping the existing service from running.
Recent BART extensions are estimated to cost over $200 million per mile. By contrast, the total costs for electrifying the existing Caltrain line, enabling it to provide service both faster and more luxurious than BART's, is between $4 million and $5 million per mile, or about one-fortieth the cost! In contrast, an upgraded Caltrain could provide as good or better level of service, be ready to run in just in three years, cost a tiny fraction of BART, and happen while existing diesel Caltrains continued to carry more and more passengers.
In the end we have to remember that the point is to carry passengers and to provide a good level of service. BART isn't the only way of doing this, as the hundreds of more Caltrain-like electric railroads in places like Japan and Switzerland show. Expanding BART is in fact the most costly and slowest way of delivering real improvements quickly. Standard rail like Caltrain also has the potential to connect seamlessly with future Dumbarton rail service, ACE service, and Union City-Fremont-San Jose rail service to San Francisco, Salinas, Stockton, Sacramento and points beyond.
