Good question. My take is:
1) If you were on a published arrival (STAR) and were cleared for the arrival initially, then further cleared to onto the ILS, then you can descend with the published arrival, which presumably would be 3000feet. Note, you must still maintain all step down height constraints as published.
2) On a radar heading the arrival is no longer in use. Therefore you cannot descend below the cleared altitude. At 4000, you need to be intercepting from at least 14nm away from glideslope. If you are being vectored inside that, ask for a lower altitude, reduce speed, configure early, be prepared to intercept from above and and above all, be prepared to ask for extra miles and go around.
3) Depending on your company, if VMC, you may ask for a visual approach whereby you can then descend at your 'leisure'. However you are not then cleared for the ILS approach, even though you will be using it as a reference.
Note, due terrain, some airports simply won't be able to descend you lower until you are clear of terrain. In this case, ATC should notify you 'standby for further descent'.
Hope that helps