Let us consider 

sol := pdsolve({diff(u(x, t), t)-(diff(v(x, t), x))+u(x, t)+v(x, t) = (1+t)*x+(x-1)*t^2, diff(v(x, t), t)-(diff(u(x, t), x))+u(x, t)+v(x, t) = (1+t)*x*t+(2*x-1)*t}, {u(0, t) = 0, u(x, 0) = 0, v(0, t) = 0, v(x, 0) = 0}, time = t, numeric, timestep = 0.1e-1, spacestep = 0.1e-1, range = 0 .. 1); 
sol:-plot3d(v(x, t), x = 0 .. 1, t = 0 .. 1);

A nice plot similar to the one produced by Mma (see the  attached pdf file pdesystem.pdf) is expected. 
The exact solutions u(x,t)=x*t,v(x,t)=x*t^2 are known

pdetest({u(x, t) = x*t, v(x, t) = x*t^2}, {diff(u(x, t), t)-(diff(v(x, t), x))+u(x, t)+v(x, t) =
(1+t)*x+(x-1)*t^2, diff(v(x, t), t)-(diff(u(x, t), x))+u(x, t)+v(x, t) = (1+t)*x*t+(2*x-1)*t});
                              {0}

But the wrong result

               module() ... end module         
Error, (in pdsolve/numeric/plot3d) unable to compute solution for t>HFloat(0.26000000000000006):
solution becomes undefined, problem may be ill posed or method may be ill suited to solution

is obtained. Also 

sol:-plot3d(v(x, t), x = 0 .. 1, t = 0 ..0.1);

The plot 

sol:-plot3d(v(x, t), x = 0 .. .5, t = 0 .. .1);

is not better.