diff --git a/tests/scripts/generate_psa_tests.py b/tests/scripts/generate_psa_tests.py
index dd5c417..5489590 100755
--- a/tests/scripts/generate_psa_tests.py
+++ b/tests/scripts/generate_psa_tests.py
@@ -516,6 +516,32 @@
         self.version = version #type: int
         self.forward = forward #type: bool
 
+    RSA_OAEP_RE = re.compile(r'PSA_ALG_RSA_OAEP\((.*)\)\Z')
+    @classmethod
+    def valid_key_size_for_algorithm(
+            cls,
+            key_type: psa_storage.Expr, bits: int,
+            alg: psa_storage.Expr
+    ) -> bool:
+        """Whether the given key type and size are valid for the algorithm.
+
+        Normally only the type and algorithm matter for compatibility, and
+        this is handled in crypto_knowledge.KeyType.can_do(). This function
+        exists to detect exceptional cases. Exceptional cases detected here
+        are not tested in OpFail and should therefore have manually written
+        test cases.
+        """
+        #pylint: disable=unused-argument
+        # OAEP requires room for two hashes plus wrapping
+        m = cls.RSA_OAEP_RE.match(alg.string)
+        if m:
+            hash_alg = m.group(1)
+            hash_length = crypto_knowledge.Algorithm.hash_length(hash_alg)
+            key_length = (bits + 7) // 8
+            # Leave enough room for at least one byte of plaintext
+            return key_length > 2 * hash_length + 2
+        return True
+
     def make_test_case(self, key: StorageTestData) -> test_case.TestCase:
         """Construct a storage format test case for the given key.
 
@@ -544,7 +570,8 @@
             # encodings of the attributes.
             # Raw data keys have no useful exercise anyway so there is no
             # loss of test coverage.
-            if key.type.string != 'PSA_KEY_TYPE_RAW_DATA':
+            if key.type.string != 'PSA_KEY_TYPE_RAW_DATA' and \
+               self.valid_key_size_for_algorithm(key.type, key.bits, key.alg):
                 flags.append('TEST_FLAG_EXERCISE')
             if 'READ_ONLY' in key.lifetime.string:
                 flags.append('TEST_FLAG_READ_ONLY')
diff --git a/tests/suites/test_suite_psa_crypto_storage_format.v0.data b/tests/suites/test_suite_psa_crypto_storage_format.v0.data
index bd7b9b4..f1f7c54 100644
--- a/tests/suites/test_suite_psa_crypto_storage_format.v0.data
+++ b/tests/suites/test_suite_psa_crypto_storage_format.v0.data
@@ -7218,7 +7218,7 @@
 
 read type: RSA_PAIR 1024-bit, RSA_OAEP(SHA_512)
 depends_on:PSA_WANT_ALG_RSA_OAEP:PSA_WANT_ALG_SHA_512:PSA_WANT_KEY_TYPE_RSA_KEY_PAIR
-key_storage_read:0x0001:PSA_KEY_TYPE_RSA_KEY_PAIR:1024:PSA_KEY_USAGE_DECRYPT | PSA_KEY_USAGE_ENCRYPT | PSA_KEY_USAGE_EXPORT:PSA_ALG_RSA_OAEP(PSA_ALG_SHA_512):0x0000:"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":"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":TEST_FLAG_EXERCISE
+key_storage_read:0x0001:PSA_KEY_TYPE_RSA_KEY_PAIR:1024:PSA_KEY_USAGE_DECRYPT | PSA_KEY_USAGE_ENCRYPT | PSA_KEY_USAGE_EXPORT:PSA_ALG_RSA_OAEP(PSA_ALG_SHA_512):0x0000:"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":"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":0
 
 read type: RSA_PAIR 1024-bit, RSA_PKCS1V15_CRYPT
 depends_on:PSA_WANT_ALG_RSA_PKCS1V15_CRYPT:PSA_WANT_KEY_TYPE_RSA_KEY_PAIR
@@ -7530,7 +7530,7 @@
 
 read type: RSA_PUB 1024-bit, RSA_OAEP(SHA_512)
 depends_on:PSA_WANT_ALG_RSA_OAEP:PSA_WANT_ALG_SHA_512:PSA_WANT_KEY_TYPE_RSA_PUBLIC_KEY
-key_storage_read:0x0001:PSA_KEY_TYPE_RSA_PUBLIC_KEY:1024:PSA_KEY_USAGE_ENCRYPT | PSA_KEY_USAGE_EXPORT:PSA_ALG_RSA_OAEP(PSA_ALG_SHA_512):0x0000:"30818902818100af057d396ee84fb75fdbb5c2b13c7fe5a654aa8aa2470b541ee1feb0b12d25c79711531249e1129628042dbbb6c120d1443524ef4c0e6e1d8956eeb2077af12349ddeee54483bc06c2c61948cd02b202e796aebd94d3a7cbf859c2c1819c324cb82b9cd34ede263a2abffe4733f077869e8660f7d6834da53d690ef7985f6bc30203010001":"505341004b455900000000000100000001400004010100000b030007000000008c00000030818902818100af057d396ee84fb75fdbb5c2b13c7fe5a654aa8aa2470b541ee1feb0b12d25c79711531249e1129628042dbbb6c120d1443524ef4c0e6e1d8956eeb2077af12349ddeee54483bc06c2c61948cd02b202e796aebd94d3a7cbf859c2c1819c324cb82b9cd34ede263a2abffe4733f077869e8660f7d6834da53d690ef7985f6bc30203010001":TEST_FLAG_EXERCISE
+key_storage_read:0x0001:PSA_KEY_TYPE_RSA_PUBLIC_KEY:1024:PSA_KEY_USAGE_ENCRYPT | PSA_KEY_USAGE_EXPORT:PSA_ALG_RSA_OAEP(PSA_ALG_SHA_512):0x0000:"30818902818100af057d396ee84fb75fdbb5c2b13c7fe5a654aa8aa2470b541ee1feb0b12d25c79711531249e1129628042dbbb6c120d1443524ef4c0e6e1d8956eeb2077af12349ddeee54483bc06c2c61948cd02b202e796aebd94d3a7cbf859c2c1819c324cb82b9cd34ede263a2abffe4733f077869e8660f7d6834da53d690ef7985f6bc30203010001":"505341004b455900000000000100000001400004010100000b030007000000008c00000030818902818100af057d396ee84fb75fdbb5c2b13c7fe5a654aa8aa2470b541ee1feb0b12d25c79711531249e1129628042dbbb6c120d1443524ef4c0e6e1d8956eeb2077af12349ddeee54483bc06c2c61948cd02b202e796aebd94d3a7cbf859c2c1819c324cb82b9cd34ede263a2abffe4733f077869e8660f7d6834da53d690ef7985f6bc30203010001":0
 
 read type: RSA_PUB 1024-bit, RSA_PKCS1V15_CRYPT
 depends_on:PSA_WANT_ALG_RSA_PKCS1V15_CRYPT:PSA_WANT_KEY_TYPE_RSA_PUBLIC_KEY