First remind that the `fitdistr` function (from the MASS package) is a
very general function that can work with nearly any distribution. 
The warnings come from non-allowed parameter values (e.g. negative
scale or shape) met during the optimisation unconstrained by default.

It seems a good idea here to try a *specific* MLE for the Weibull
distribution. A quite well-known fact is that the ML estimation of the
two-parameter Weibull can be rely on a concentration of the
log-likelihood, leading to an easier *one-dimensional*
optimisation. Moreover, the concentrated log-likelihood is concave, so
there is a unique ML estimate.

The problem here is that the log-likelihood is quite flat near the
optimum, so different optimisations lead to different results as
reported by @Glen_b. Moreover, the data scaling is prone to numerical
problems. After rescaling, similar results are obtained with or without
concentration.  A general practical finding about MLE is
that using poorly scaled data can be enough to ruin the estimation.

    > library(Renext)            ## for concentrated log-lik
    > try(fweibull(Y))           ## error (numerical pb with information matrix)
    > fit <- fweibull(Y / 1000)  ## works
    > ## set parameters and logLik back to original scale
    > fit$est * c(1, 1000)
          shape       scale 
       2.126225 1563.094460

    > fit$sd * c(1, 1000)
          shape       scale 
      0.2444308 114.1293266
 
    > fit$loglik - length(Y) * log(1000)
    [1] -362.2237

    > library(MASS)
    > ## set parameters and logLik back to original scale
    > fit2 <- fitdistr(Y / 1000, "weibull")
    > fit2$est * c(1, 1000)
          shape       scale 
       2.126231 1563.095165 

    > fit2$sd * c(1, 1000)
          shape       scale 
      0.2288605 114.9071653 

    > fit2$loglik - length(Y) * log(1000)
    [1] -362.2237