[HTC One X] ninelo:
After some time of testing I came to conclusion that the reason was the error "EROFS (Read-only file system)".

A few examples:

Line 69: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 87: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 171: E/DropBoxManagerService( 358): java.io.FileNotFoundException: /data/system/dropbox/drop272.tmp: open failed: EROFS (Read-only file system)
Line 187: E/DropBoxManagerService( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 279: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 297: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 311: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 329: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 347: E/DropBoxManagerService( 358): java.io.FileNotFoundException: /data/system/dropbox/drop270.tmp: open failed: EROFS (Read-only file system)
Line 361: E/DropBoxManagerService( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 395: W/BatteryStats( 358): java.io.FileNotFoundException: /data/system/batterystats.bin.tmp: open failed: EROFS (Read-only file system)
Line 407: W/BatteryStats( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 443: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 461: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 485: E/DropBoxManagerService( 358): java.io.FileNotFoundException: /data/system/dropbox/drop278.tmp: open failed: EROFS (Read-only file system)
Line 499: E/DropBoxManagerService( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 2201: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 2219: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 2233: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 2251: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 2413: E/DropBoxManagerService( 358): java.io.FileNotFoundException: /data/system/dropbox/drop285.tmp: open failed: EROFS (Read-only file system)
Line 2429: E/DropBoxManagerService( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 2527: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 2545: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 2559: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 2577: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 3039: E/libEGL ( 9262): error creating cache file /data/data/com.htc.Weather/cache/com.android.opengl.shaders_cache: Read-only file system (30)
Line 3195: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 3213: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 3231: W/SqliteDatabaseCpp( 9327): getJournalMode: errno = 30, error message = Read-only file system
Line 3235: E/SqliteDatabaseCpp( 9327): native_setLocale: errno = 30, error message = Read-only file system
Line 3545: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 3563: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 3595: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 3613: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 4039: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 4057: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 4087: W/ActivityManager( 358): java.io.FileNotFoundException: /data/anr/traces.txt: open failed: EROFS (Read-only file system)
Line 4105: W/ActivityManager( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 5189: E/DropBoxManagerService( 358): java.io.FileNotFoundException: /data/system/dropbox/drop179.tmp: open failed: EROFS (Read-only file system)
Line 5205: E/DropBoxManagerService( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 5729: W/BatteryStats( 358): java.io.FileNotFoundException: /data/system/batterystats.bin.tmp: open failed: EROFS (Read-only file system)
Line 5741: W/BatteryStats( 358): Caused by: libcore.io.ErrnoException: open failed: EROFS (Read-only file system)
Line 7275: W/SqliteDatabaseCpp( 9677): getJournalMode: errno = 30, error message = Read-only file system
Line 7279: E/SqliteDatabaseCpp( 9677): native_setLocale: errno = 30, error message = Read-only file system

It remains to find a solution...