A corporate jet Captain reported both the and the Dispatcher incorrectly filed for an alternate that had no TAF weather reporting capability.

Narrative

The forecast weather at our planned destination; ACK; was pretty good when I arrived at the airport for the trip and did not require an alternate. Because I was on Day 1 of my tour and airlining in with an arrival time fairly close to trip departure time; Dispatch filed the flight plan for me. Shortly before departure time I called Dispatch for updated weather and was surprised to hear that the METAR was reporting low visibility; less than one mile. Because the flight was less than an hour I decided to file an alternate based on the METAR. Because Dispatch had filed the original flight plan; I asked the Dispatcher to re-file with an alternate. A few minutes later a Dispatcher called me to confirm that I wanted to file a new flight plan with an alternate. I said yes and suggested BOS. The Dispatcher than said; 'we were thinking New Bedford.' I heard 'Bedford;' and so I thought he was saying that BED was the new filed alternate. I thought BED was acceptable because I wrongly assumed BED had a TAF. The Dispatcher told me he'd fax the new flight plan and weather to me and we hung up. I got the new paperwork just as the passengers arrived. I scanned the flight plan to see that the alternate was on there but didn't register in my mind that it was EWB and that it didn't have its own TAF. It wasn't until later; during the flight; that I looked at the weather again and realized that we had filed an alternate with no weather forecast. So this was a two-part goof. First of all; I misunderstood the Dispatcher when he told me the chosen alternate. I heard 'Bedford' when he actually said 'New Bedford.' I was mistaken in assuming BED had a TAF. Even if the filed airport had been BED; it still wouldn't have been an acceptable alternate because it doesn't have a TAF either. And secondly; I should have caught that the actual alternate; EWB; didn't have forecast weather when I picked up the fax with the weather. I attribute this primarily to the poor timing of picking up the weather fax (the computers in the FBO were unavailable) as the passengers were arriving. I let the passengers distract me from giving the weather page my full attention. A secondary factor was my misunderstanding about BED being the filed alternate and thinking that it had a TAF. I had checked the BOS TAF (which was very good) before making the call to Dispatch about filing an alternate and since BED and BOS are so close I just had it in my mind that we were in the clear. So there were multiple incorrect assumptions involved on my part.