lme4, error in mer_finalize(ans)

Using lmer() on my data results in an error. The problem, I think, is my
model specification. However, lm() works ok.
I recreated this error with a more simple dataset. (See code below.)

# word and letter recognition data
# two within factors:
# word length: 4, 5, 6 letters
# letter position: 1-4 (in 4-letter words), 1-5 (in 5-letter words), 1-6 (in
6-letter words)
# one dependent variable:
# reaction time

# first fit a linear model with letter position nested in word length
lm(rt ~ length + length:pos, data=df)

# fit a mixed effects model, with subj (participant) as random effect
lmer(rt ~ length + length:pos + (1 | subj), data=df)

Using lmer() results in an error: Error in mer_finalize(ans) : Downdated X'X
is not positive definite, 13. I don't experience any problems using lm().
Does anyone know where things go wrong?

That, admittedly obscure, error message relates to the fixed-effects
specification rt ~ length + length:pos being rank deficient.  If you
look at the summary of the linear model fit you will see that there
are 3 coefficients that are not determined because of singularities.
The lm function detects the singularities and fits a lower-rank model.
 The lmer function is not as sophisticated.  It just detects the
singularities and quits.

The length and the position are confounded.

xtabs(~ len + pos, df)

pos
len 1 2 3 4 5 6
  4 2 2 2 2 0 0
  5 2 2 2 2 2 0
  6 2 2 2 2 2 2

(By the way, I changed the name of the length variable to len as
typing "length" makes me expect the function called length.)

Even when you remove this confounding by creating the len:pos
interaction separately as a factor, you will still get singularities
because there is only one len:pos combination for len = 6.

You will need to think of a way of parameterizing the fixed effects
without the singularities.  You can check for singularities in the
summary of the lm fit.