Feynmann certainly leaves the impression that one can.
Certainly the story Feynman tells in volume II section 1-5 is
open to misinterpretation. He says his moving charges have a
magnetic field (in the lab frame). That's true as far as it
goes, but he fails to mention that they also have an electric
field. This *particular* combination of E *and* B in the lab
frame will have no B component in another artfully-chosen frame.
Leaving behind this particular example and this particular
authority, it is better to _understand_ the situation.
By way of warm-up exercise, consider the relatively well-known
fact that a timelike 4-vector will look timelike to all observers.
Similarly a spacelike 4-vector will look spacelike to all observers.
Last but not least a null 4-vector will look null to all observers.
You can prove this in your head in less time than it takes to tell
about it, by computing the dot product of the vector with itself.
Any dot product produces a Lorentz invariant scalar.
This particular dot product, namely the dot product of a vector
with itself, is what I call the _gorm_ of the vector. In Euclidean
space, it is identical to the norm squared, but in Minkowski space,
the gorm is not generally the square of anything, so we just call
it the gorm.
Now the fun part is that in spacetime, the electromagnetic field
is represented by a bivector (F). And it turns out that gorm is
defined for bivectors, too. In particular, the gorm of any bivector
F is the scalar part of F F~ i.e. the scalar part of the product
of F with the reverse of F. If you work out what F is in terms
of some frame's E-field and B-field, you find that the gorm of F
is (cB)^2 - E^2. That means that if you change frames, E will be
different and B will be different (in general), but (cB)^2 - E^2
will be the same, since it is a Lorentz invariant scalar.
To summarize: We know that pure B cannot turn into pure E for more
or less the same reason that a spacelike vector cannot turn into a
timelike vector. In an important sense, B is the spacelike part of
F, while E is the timelike part.