The Hungarian Rail Company, MÁV, is at least trying to improve its bicycle carrying services. In June, it posted a couple of PDF tables on its special cycling page http://www.mav-start.hu/mavstart/kerek_par.php listing all the trains that have dedicated cars for bicycles. The regular timetable page http://www.mav-start.hu/ indicates whether or not there's a bike car on a given train when you click the departure time for details. But these tables list all the trains with bicycle cars into four neat pages, which you can print out and take with you so you don't get stuck in the countryside with no train home.
I've taken my bike on MÁV several times, and the experiences have been mixed. The bike cars can make for a convenient, secure place to put your bike -- provided they aren't too crowded. On trains without dedicated bike stowage, you have the option of putting your bike at the end of a car, but this isn't very comfortable or secure, and we've run into conductors who look at this with a very jaundiced eye despite official policy that permits two bikes at each end of every car.
One of my worst experiences in this vein was in the summer of 2004 when my wife and I went for a weekend ride around Fertod Lake on the Austrian border. Going there was no problem, but on the return trip to Budapest on Sunday evening, we got stuck at the Sopron station for several hours, after a ticket clerk refused to let us board a mostly empty train on grounds that there was no room for bikes. We managed to argue our way onto a train late in the evening, and were bitched at by a bike-hating conductor the whole way back. We wrote a letter of complaint, which got a dismissive response, and later I vented in a long letter to the editor of the Budapest Sun, which can still be accessed on-line http://www.budapestsun.com/cikk.php?id=16094.
But it would seem things are getting better. MÁV started a long-term development plan for bicycle service in 2006, and now there are bicycle cars on trains going all over the country. For some reason, MÁV doesn't allow bikes on the Intercities. They're not allowed on the ends of passenger cars and, at least until recently, the Intercities haven't carried dedicated bike cards. However, this apparently is also changing, and now there are a couple Intercity routes -- including the one between Budpaest and Sopron -- that have bike cars. There should be bike cars on all Intercities in my opinion. I guarantee that the ones to Balaton would be oversubscribed from day one.
I base this on our experience in mid-July on a Sunday evening train returning from Balatonfüred. The bike car was so full that people were literally stacking bicycles on top of each other. It was a wonder no one got crushed. We were lucky enough to have gotten on at Füred when there was room to hang and lock up our bikes and get a seat. After Füred, the bikes started stacking up, and the cyclists themselves had to stand in the car for the duration to help unstack and restack bikes at every stop. There was no conductor in sight, and if there had been, it would have been impossible for him to even get into the bike car, let alone count bikes and collect fares.
Based on my experiences, I would advise anyone travelling by bike to plan ahead for both outgoing and returning trips to ensure there is a bike car. On trains without bike cars, you are still allowed to put two bikes at each end of every car, but this is not a comfortable arrangement, and it's difficult to lock up, as well. Also, if you're taking a trip at a peak time, try to arrange your departure as close to the train's origin as possible, so that you can board before the bike car gets too crowded.