(note: using language that accepts the claims of the paper for simplicity -- I remain skeptical)
For what it's worth, superconductors have a shared budget[1] of (magnetic field, temperature, current). At 25°C, the material is near its critical temperature, so its current-carrying capacity is necessarily diminished. At a lower temperature, the film should be able to carry more current.
That said, 250mA is plenty of current if you're interested in making a superconducting CPU.
[1] http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/scbc2.html
edit after reading the paper: they claim an extraordinarily high critical temperature of ~126°C. You can see the temperature dependence in Figure 1e; they're much further from the critical temperature than I expected, and at room temperature, a little cooling appears to go a long way. I'm eager to see an attempt to reproduce this result. That said, the material is essentially a 2d molecule -- we've been hyped on graphene for decades, and have yet to see it integrated into a scalable process.