diff --git a/Lib/test/pickletester.py b/Lib/test/pickletester.py
index 1f59ab2..a2bea60 100644
--- a/Lib/test/pickletester.py
+++ b/Lib/test/pickletester.py
@@ -21,8 +21,6 @@
 # kind of outer loop.
 protocols = range(pickle.HIGHEST_PROTOCOL + 1)
 
-ascii_char_size = 1
-
 
 # Return True if opcode code appears in the pickle, else False.
 def opcode_in_pickle(code, pickle):
@@ -1643,7 +1641,7 @@
 
     # Binary protocols can serialize longs of up to 2GB-1
 
-    @bigmemtest(size=_2G, memuse=1 + 1, dry_run=False)
+    @bigmemtest(size=_2G, memuse=3.6, dry_run=False)
     def test_huge_long_32b(self, size):
         data = 1 << (8 * size)
         try:
@@ -1660,7 +1658,7 @@
     # (older protocols don't have a dedicated opcode for bytes and are
     # too inefficient)
 
-    @bigmemtest(size=_2G, memuse=1 + 1, dry_run=False)
+    @bigmemtest(size=_2G, memuse=2.5, dry_run=False)
     def test_huge_bytes_32b(self, size):
         data = b"abcd" * (size // 4)
         try:
@@ -1681,7 +1679,7 @@
         finally:
             data = None
 
-    @bigmemtest(size=_4G, memuse=1 + 1, dry_run=False)
+    @bigmemtest(size=_4G, memuse=2.5, dry_run=False)
     def test_huge_bytes_64b(self, size):
         data = b"acbd" * (size // 4)
         try:
@@ -1711,7 +1709,7 @@
     # All protocols use 1-byte per printable ASCII character; we add another
     # byte because the encoded form has to be copied into the internal buffer.
 
-    @bigmemtest(size=_2G, memuse=2 + ascii_char_size, dry_run=False)
+    @bigmemtest(size=_2G, memuse=8, dry_run=False)
     def test_huge_str_32b(self, size):
         data = "abcd" * (size // 4)
         try:
@@ -1738,7 +1736,7 @@
     # of utf-8 encoded unicode. BINUNICODE8 (protocol 4) supports these huge
     # unicode strings however.
 
-    @bigmemtest(size=_4G, memuse=2 + ascii_char_size, dry_run=False)
+    @bigmemtest(size=_4G, memuse=8, dry_run=False)
     def test_huge_str_64b(self, size):
         data = "abcd" * (size // 4)
         try:
diff --git a/Lib/test/test_bz2.py b/Lib/test/test_bz2.py
index cd58f31..26703f5 100644
--- a/Lib/test/test_bz2.py
+++ b/Lib/test/test_bz2.py
@@ -686,7 +686,7 @@
         self.assertRaises(EOFError, bz2d.decompress, b"anything")
         self.assertRaises(EOFError, bz2d.decompress, b"")
 
-    @bigmemtest(size=_4G + 100, memuse=3)
+    @bigmemtest(size=_4G + 100, memuse=3.3)
     def testDecompress4G(self, size):
         # "Test BZ2Decompressor.decompress() with >4GiB input"
         blocksize = 10 * 1024 * 1024
diff --git a/Lib/test/test_hashlib.py b/Lib/test/test_hashlib.py
index 179e771..36a98cb 100644
--- a/Lib/test/test_hashlib.py
+++ b/Lib/test/test_hashlib.py
@@ -256,21 +256,15 @@
                    b'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789',
                    'd174ab98d277d9f5a5611c2c9f419d9f')
 
-    @bigmemtest(size=_4G + 5, memuse=1)
+    @unittest.skipIf(sys.maxsize < _4G + 5, 'test cannot run on 32-bit systems')
+    @bigmemtest(size=_4G + 5, memuse=1, dry_run=False)
     def test_case_md5_huge(self, size):
-        if size == _4G + 5:
-            try:
-                self.check('md5', b'A'*size, 'c9af2dff37468ce5dfee8f2cfc0a9c6d')
-            except OverflowError:
-                pass # 32-bit arch
+        self.check('md5', b'A'*size, 'c9af2dff37468ce5dfee8f2cfc0a9c6d')
 
-    @bigmemtest(size=_4G - 1, memuse=1)
+    @unittest.skipIf(sys.maxsize < _4G - 1, 'test cannot run on 32-bit systems')
+    @bigmemtest(size=_4G - 1, memuse=1, dry_run=False)
     def test_case_md5_uintmax(self, size):
-        if size == _4G - 1:
-            try:
-                self.check('md5', b'A'*size, '28138d306ff1b8281f1a9067e1a1a2b3')
-            except OverflowError:
-                pass # 32-bit arch
+        self.check('md5', b'A'*size, '28138d306ff1b8281f1a9067e1a1a2b3')
 
     # use the three examples from Federal Information Processing Standards
     # Publication 180-1, Secure Hash Standard,  1995 April 17
@@ -379,14 +373,11 @@
           "e718483d0ce769644e2e42c7bc15b4638e1f98b13b2044285632a803afa973eb"+
           "de0ff244877ea60a4cb0432ce577c31beb009c5c2c49aa2e4eadb217ad8cc09b")
 
-    @bigmemtest(size=_4G + 5, memuse=1)
+    @unittest.skipIf(sys.maxsize < _4G + 5, 'test cannot run on 32-bit systems')
+    @bigmemtest(size=_4G + 5, memuse=1, dry_run=False)
     def test_case_sha3_224_huge(self, size):
-        if size == _4G + 5:
-            try:
-                self.check('sha3_224', b'A'*size,
-                           '58ef60057c9dddb6a87477e9ace5a26f0d9db01881cf9b10a9f8c224')
-            except OverflowError:
-                pass # 32-bit arch
+        self.check('sha3_224', b'A'*size,
+                   '58ef60057c9dddb6a87477e9ace5a26f0d9db01881cf9b10a9f8c224')
 
     def test_gil(self):
         # Check things work fine with an input larger than the size required
diff --git a/Lib/test/test_marshal.py b/Lib/test/test_marshal.py
index 255eb87..e477c46 100644
--- a/Lib/test/test_marshal.py
+++ b/Lib/test/test_marshal.py
@@ -288,19 +288,19 @@
     def check_unmarshallable(self, data):
         self.assertRaises(ValueError, marshal.dump, data, NullWriter())
 
-    @support.bigmemtest(size=LARGE_SIZE, memuse=1, dry_run=False)
+    @support.bigmemtest(size=LARGE_SIZE, memuse=2, dry_run=False)
     def test_bytes(self, size):
         self.check_unmarshallable(b'x' * size)
 
-    @support.bigmemtest(size=LARGE_SIZE, memuse=1, dry_run=False)
+    @support.bigmemtest(size=LARGE_SIZE, memuse=2, dry_run=False)
     def test_str(self, size):
         self.check_unmarshallable('x' * size)
 
-    @support.bigmemtest(size=LARGE_SIZE, memuse=pointer_size, dry_run=False)
+    @support.bigmemtest(size=LARGE_SIZE, memuse=pointer_size + 1, dry_run=False)
     def test_tuple(self, size):
         self.check_unmarshallable((None,) * size)
 
-    @support.bigmemtest(size=LARGE_SIZE, memuse=pointer_size, dry_run=False)
+    @support.bigmemtest(size=LARGE_SIZE, memuse=pointer_size + 1, dry_run=False)
     def test_list(self, size):
         self.check_unmarshallable([None] * size)
 
@@ -316,7 +316,7 @@
     def test_frozenset(self, size):
         self.check_unmarshallable(frozenset(range(size)))
 
-    @support.bigmemtest(size=LARGE_SIZE, memuse=1, dry_run=False)
+    @support.bigmemtest(size=LARGE_SIZE, memuse=2, dry_run=False)
     def test_bytearray(self, size):
         self.check_unmarshallable(bytearray(size))
 
diff --git a/Lib/test/test_xml_etree_c.py b/Lib/test/test_xml_etree_c.py
index b3ff7ae..816aa86 100644
--- a/Lib/test/test_xml_etree_c.py
+++ b/Lib/test/test_xml_etree_c.py
@@ -13,10 +13,8 @@
 
 class MiscTests(unittest.TestCase):
     # Issue #8651.
-    @support.bigmemtest(size=support._2G + 100, memuse=1)
+    @support.bigmemtest(size=support._2G + 100, memuse=1, dry_run=False)
     def test_length_overflow(self, size):
-        if size < support._2G + 100:
-            self.skipTest("not enough free memory, need at least 2 GB")
         data = b'x' * size
         parser = cET.XMLParser()
         try: