Wikipedia has several interesting articles on this topic, starting here:
http://en.wikipedia.org/wiki/Nuclear_weapon_design
The "access to the fissile material" part is by far the most unrealistic to me. How one gets that is the biggest handwave needed. Any country that develops uranium-based bombs spends huge amounts of money refining the U-235 isotope from the ore (which contains mostly nonradioactive U-238) to the purity (something like 75 to 95 percent) needed to make a bomb. I know less about plutonium, except that it is also very expensive and difficult to produce enough of it for a bomb. Refining the material is also the part that appears to take the most technology. This, probably as much as all other factors put together, is why there aren't more countries in the
Nuclear Club.
At its crudest, what you need is enough fissible material, enough explosives to drive the material together, a strong enough vessel for this to happen in, and a neutron source.
The neutron source is probably the hardest to come by. It more or less requires a reactor or a particle accellerator to make.
So I would say no.
Neutrons are regularly generated by random radioactive decay in the fissile material. It even has a name when a decay-generated neutron causes a chain reaction "too early" while the material is being assembled (before the material comes together enough to make a really good explosion),
predetonation. But even a bomb that experiences predetonation will likely make a big, ugly explosion with several or many times the bomb's weight in TNT.
You don't even need explosives. With enough fissile material (and the right knowledge of critical mass) you could create a nuke in cave-man days. All you have to do is put two sub-critical masses close enough together.
You get some really bad radiation doing that, but it won't actually explode. The material will push itself apart as it becomes critical. It
has happened accidentially.
I recall such a homemade nuke in Rudy Rucker's "The Sex Sphere" where the two equally-sized subcritical parts were put together by hand, by "playing the cymbals." I knew less about it then (because there wasn't a Wikipedia to learn so much about this), but it looks unrealistic now, at least as far as making a big explosion. For making some heat, blue glow and a lot of deadly radiation, yeah, it'll sure do that.
Explosives wouldn't actually help at all, because the timing has to be nanosecond precise when using that method.
For the implosion method, yes (maybe not NANOsecond precise), but for the
gun method any explosives used to put the parts together faster helps make a bigger yield. The more powerful the explosives, the better it helps.
The implosion method, on the other hand, does require carefully shaped and timed explosives to get a decent yield.