No, CF=0 occurences seem to be happen frequently and uniformely distributed like valid results at ~1/65536, not clustered. Under a minute-long all-core load CF=0 always produces zero, but that's to be expected according to the manual.

Here are some stats:

    Rounds (N): 1000000000
    Failed (F): 15312
    Valid  (V): 999984688
    N/65536: 15258.789
    V/65536: 15258.555
    Failed, result was zero: 15312
    Failed, result non-zero: 0
    Bucket value for      0: 15312
    Bucket value for      1: 15290
    Bucket value for  65535: 15223
    Min bucket value: 14670
    Max bucket value: 15835

I used this C program to collect them:

    #include <stdio.h>
    #include <stdint.h>
    #include <stdbool.h>

    const size_t N = 1000000000; // 1e9

    struct rdrand16_result {
        uint16_t n;
        bool ok;
    };

    static inline struct rdrand16_result rdrand16()
    {
        struct rdrand16_result result;
        __asm__ __volatile__( "rdrand %0" : "=r" (result.n), "=@ccc" (result.ok) );
        return result;
    }

    int main()
    {
        size_t buckets[0xFFFF + 1] = { 0 };
        size_t notok = 0, notok_zero = 0, notok_nonz = 0;
        for (size_t i = 0; i < N; ++i) {
            struct rdrand16_result result = rdrand16();
            ++buckets[result.n];
            if (! result.ok) {
                ++notok;
                notok_zero += result.n == 0;
                notok_nonz += result.n != 0;
            }
        }
        size_t max = 0, min = N;
        for (size_t i = 0; i <= 0xFFFF; ++i) {
            size_t n = buckets[i];
            min = n < min ? n : min;
            max = n > max ? n : max;
        }
        printf("Rounds (N): %zu\n", N);
        printf("Failed (F): %zu\n", notok);
        printf("Valid  (V): %zu\n", N - notok);
        printf("N/65536: %.3f\n", (double)N / 65536);
        printf("V/65536: %.3f\n", (double)(N - notok) / 65536);
        printf("Failed, result was zero: %zu\n", notok_zero);
        printf("Failed, result non-zero: %zu\n", notok_nonz);
        printf("Bucket value for      0: %zu\n", buckets[0]);
        printf("Bucket value for      1: %zu\n", buckets[1]);
        printf("Bucket value for  65535: %zu\n", buckets[0xFFFF]);
        printf("Min bucket value: %zu\n", min);
        printf("Max bucket value: %zu\n", max);
        return 0;
    }