Tests:

java_io/File/FileAttributes/FATest4
java_io/File/FileAttributes/FATest6

check for the inability to read/write files that have read/write permissions turned off. However if running the test as the root user these tests fail because root can read/write regardless of permission bits.

The tests should check for being run by root and simply report success in that case.
One more test has same problem:
java_nio/file/TestDelete01
These also seem to have similar issues:

   api/java_io/File/descriptions_setPermissions
   api/java_io/File/index_FilePermissions

Edited to add: these are JCK tests so not in the same category as the original tests
JCK tests mentioned above fails on solaris and linux on 6.0 fcs, on jdk7, etc.

Tests are absolutely correct according to evaluation from Arkadiy Sutchilin, so it seems that this is java RI defect. Redispatching CR to Java product.

To reproduce the jck failure, install jdk and jck of corresponding versions. Then use this simple script, after modifying path to java and jck:
#!/bin/bash

cat /etc/*ele*

whoami

/export/jdk/jdk1.6.0/bin/java -showversion -cp "/set/stt/jck_promotions/6b/latest/binaries/JCK-runtime-6b/classes" javasoft.sqe.tests.api.java.io.File.setPermissionsTests -TestCaseID ALL -TestURL file:////set/stt/jck_promotions/6b/latest/binaries/JCK-runtime-6b/tests/api/java_io/File/descriptions.html -TestWorkDirURL file:///export/JCK

/export/jdk/jdk1.6.0/bin/java -showversion -cp "/set/stt/jck_promotions/6b/latest/binaries/JCK-runtime-6b/classes" javasoft.sqe.tests.api.java.io.File.FilePermissionsTests -TestCaseID ALL -TestURL file:////set/stt/jck_promotions/6b/latest/binaries/JCK-runtime-6b/tests/api/java_io/File/index.html -TestWorkDirURL file:///export/JCK

The output:

                   Oracle Solaris 10 9/10 s10s_u9wos_14a SPARC
     Copyright (c) 2010, Oracle and/or its affiliates. All rights reserved.
                            Assembled 11 August 2010
root
java version "1.6.0"
Java(TM) SE Runtime Environment (build 1.6.0-b105)
Java HotSpot(TM) Client VM (build 1.6.0-b105, mixed mode, sharing)

File0061: Failed. setReadOnly operation succeed but canWrite method returns true
File0062: Passed. OKAY.
File0063: Failed. Readonly file was modified
File0064: Passed. OKAY.
File0065: Passed. OKAY
File0066: Passed. OKAY
File0067: Passed. OKAY
File0069: Passed. OKAY
File0070: Passed. OKAY
STATUS:Failed.test cases: 9; passed: 7; failed: 2; first test case failure: File0061
java version "1.6.0"
Java(TM) SE Runtime Environment (build 1.6.0-b105)
Java HotSpot(TM) Client VM (build 1.6.0-b105, mixed mode, sharing)

executable0001: Failed. canExecute() does not correspond to setExecutable()
executable0002: Passed. OKAY.
executable0003: Failed. canExecute() does not correspond to setExecutable()
executable0004: Passed. OKAY.
executable0005: Passed. OKAY.
executable0006: Passed. OKAY.
readable0001: Failed. canRead() does not correspond to setReadable()
readable0002: Passed. OKAY.
readable0003: Failed. canRead() does not correspond to setReadable()
readable0004: Passed. OKAY.
readable0005: Passed. OKAY.
readable0006: Passed. OKAY.
writable0001: Failed. setWritable() doesn't work properly
writable0002: Passed. OKAY.
writable0003: Failed. setWritable() doesn't work properly
writable0004: Passed. OKAY.
writable0005: Passed. OKAY.
writable0006: Passed. OKAY.
canExecute() returned incorrect value: true, expected: false
canExecute() returned incorrect value: true, expected: false, ownerOnly: true
canExecute() returned incorrect value: true, expected: false, ownerOnly: false
canRead() returned incorrect value: true, expected: false
canRead() returned incorrect value: true, expected: false, ownerOnly: true
canRead() returned incorrect value: true, expected: false, ownerOnly: false
canWrite() returned incorrect value: true, expected: false
canWrite() returned incorrect value: true, expected: false, ownerOnly: true
canWrite() returned incorrect value: true, expected: false, ownerOnly: false
STATUS:Failed.test cases: 18; passed: 12; failed: 6; first test case failure: executable0001