diff --git a/numpy/random/_generator.pyx b/numpy/random/_generator.pyx
index d7c1879e7aee..3eb5876ecd46 100644
--- a/numpy/random/_generator.pyx
+++ b/numpy/random/_generator.pyx
@@ -15,6 +15,7 @@ from numpy.core.multiarray import normalize_axis_index
 
 from .c_distributions cimport *
 from libc cimport string
+from libc.math cimport sqrt
 from libc.stdint cimport (uint8_t, uint16_t, uint32_t, uint64_t,
                           int32_t, int64_t, INT64_MAX, SIZE_MAX)
 from ._bounded_integers cimport (_rand_bool, _rand_int32, _rand_int64,
@@ -2997,6 +2998,22 @@ cdef class Generator:
         then the probability distribution of the number of non-"1"s that
         appear before the third "1" is a negative binomial distribution.
 
+        Because this method internally calls ``Generator.poisson`` with an
+        intermediate random value, a ValueError is raised when the choice of 
+        :math:`n` and :math:`p` would result in the mean + 10 sigma of the sampled
+        intermediate distribution exceeding the max acceptable value of the 
+        ``Generator.poisson`` method. This happens when :math:`p` is too low 
+        (a lot of failures happen for every success) and :math:`n` is too big (
+        a lot of sucesses are allowed).
+        Therefore, the :math:`n` and :math:`p` values must satisfy the constraint:
+
+        .. math:: n\\frac{1-p}{p}+10n\\sqrt{n}\\frac{1-p}{p}<2^{63}-1-10\\sqrt{2^{63}-1},
+
+        Where the left side of the equation is the derived mean + 10 sigma of
+        a sample from the gamma distribution internally used as the :math:`lam`
+        parameter of a poisson sample, and the right side of the equation is
+        the constraint for maximum value of :math:`lam` in ``Generator.poisson``.
+
         References
         ----------
         .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
@@ -3021,9 +3038,41 @@ cdef class Generator:
         ...    print(i, "wells drilled, probability of one success =", probability)
 
         """
+
+        cdef bint is_scalar = True
+        cdef double *_dn
+        cdef double *_dp
+        cdef double _dmax_lam
+
+        p_arr = <np.ndarray>np.PyArray_FROM_OTF(p, np.NPY_DOUBLE, np.NPY_ALIGNED)
+        is_scalar = is_scalar and np.PyArray_NDIM(p_arr) == 0
+        n_arr = <np.ndarray>np.PyArray_FROM_OTF(n, np.NPY_DOUBLE, np.NPY_ALIGNED)
+        is_scalar = is_scalar and np.PyArray_NDIM(n_arr) == 0
+
+        if not is_scalar:
+            check_array_constraint(n_arr, 'n', CONS_POSITIVE_NOT_NAN)
+            check_array_constraint(p_arr, 'p', CONS_BOUNDED_GT_0_1)
+            # Check that the choice of negative_binomial parameters won't result in a
+            # call to the poisson distribution function with a value of lam too large.
+            max_lam_arr = (1 - p_arr) / p_arr * (n_arr + 10 * np.sqrt(n_arr))
+            if np.any(np.greater(max_lam_arr, POISSON_LAM_MAX)):
+                raise ValueError("n too large or p too small, see Generator.negative_binomial Notes")
+
+        else:
+            _dn = <double*>np.PyArray_DATA(n_arr)
+            _dp = <double*>np.PyArray_DATA(p_arr)
+
+            check_constraint(_dn[0], 'n', CONS_POSITIVE_NOT_NAN)
+            check_constraint(_dp[0], 'p', CONS_BOUNDED_GT_0_1)
+            # Check that the choice of negative_binomial parameters won't result in a
+            # call to the poisson distribution function with a value of lam too large.
+            _dmax_lam = (1 - _dp[0]) / _dp[0] * (_dn[0] + 10 * sqrt(_dn[0]))
+            if _dmax_lam > POISSON_LAM_MAX:
+                raise ValueError("n too large or p too small, see Generator.negative_binomial Notes")
+
         return disc(&random_negative_binomial, &self._bitgen, size, self.lock, 2, 0,
-                    n, 'n', CONS_POSITIVE_NOT_NAN,
-                    p, 'p', CONS_BOUNDED_GT_0_1,
+                    n_arr, 'n', CONS_NONE,
+                    p_arr, 'p', CONS_NONE,
                     0.0, '', CONS_NONE)
 
     def poisson(self, lam=1.0, size=None):
diff --git a/numpy/random/tests/test_generator_mt19937.py b/numpy/random/tests/test_generator_mt19937.py
index 7c61038a4f0d..3ccb9103c5c9 100644
--- a/numpy/random/tests/test_generator_mt19937.py
+++ b/numpy/random/tests/test_generator_mt19937.py
@@ -1485,6 +1485,13 @@ def test_negative_binomial_p0_exception(self):
         with assert_raises(ValueError):
             x = random.negative_binomial(1, 0)
 
+    def test_negative_binomial_invalid_p_n_combination(self):
+        # Verify that values of p and n that would result in an overflow
+        # or infinite loop raise an exception.
+        with np.errstate(invalid='ignore'):
+            assert_raises(ValueError, random.negative_binomial, 2**62, 0.1)
+            assert_raises(ValueError, random.negative_binomial, [2**62], [0.1])
+
     def test_noncentral_chisquare(self):
         random = Generator(MT19937(self.seed))
         actual = random.noncentral_chisquare(df=5, nonc=5, size=(3, 2))