Possible errors (detectable during initialisation, i.e. only applicable to new() methods):

• Failure to open random device (e.g. /dev/urandom)
• Insufficient entropy (system not ready)

Possible errors (at any time, thus affecting new(), next_* and try_fill()):

• Interrupt by user (Ctrl+C)
• Hardware/external generator failure or disconnection
• PRNG cycle
• "Entropy exhaustion": PRNGs with artificially limited output (hypothetical and not needed even for periodic reseeding; suggested here)

Proposed error kinds:

/// Error kind which can be matched over.
#[derive(PartialEq, Eq, Debug, Copy, Clone)]
pub enum ErrorKind {
    /// Permanent failure: likely not recoverable without user action.
    Unavailable,
    /// Temporary failure: recommended to retry a few times, but may also be
    /// irrecoverable.
    Transient,
    /// Not ready yet: recommended to try again a little later.
    NotReady,
    /// Uncategorised error
    Other,
    // no hidden members: allow exclusive match
}

References:

• Summary by @pitdicker — touches on most of these sources
• Should PRNGs detect and report cycle errors? See discussion starting here.