Old European hydronymy
The Old European hydronymy is an apparently uniform network of recurring river names which spans most of Central and Western Europe. This network was discovered by the German linguist Hans Krahe, who coined the term, in the middle of the 20th century (see Krahe, H., Die Struktur der alteuropäischen Hydronymie (1963)). It is, however, controversial whether this is a real phenomenon or just a case of chance resemblance, and, if real, which language family is to be held "responsible" for these names. The matter is made difficult by the fact that the original meanings of the names are unknown, which leaves much space for speculation.
Theories about the Old European hydronymy
Hans Krahe ascribed the Old European hydronymy to an Indo-European language which, according to him, was the common ancestor of Italic, Celtic, Germanic, Baltic and possibly Slavic. His evidence consists of proposed Indo-European etymologies for the river names which consist of verbs, adjectvies and nouns with meanings which are, for the most part, appropriate to the description of natural watercourses.
However, there are two problems with this hypothesis. First, these languages are not characterized by any shared innovation and do not seem to form a valid node in the Indo-European family tree; second, the "Old European" river names, while seemingly having Indo-European etymologies, do not reflect the characteristic sound changes of the languages in which they are found and are thus more likely to be borrowed rather than inherited. For example, the Old European river names show a predominant /a/-vocalism which looks quite un-Indo-European, while the vowels /e/ and /o/ which are frequent in Indo-European appear to have been absent, or where they occur, appear to be of secondary origin.
Schmid's revision of Krahe's theory
Another German linguist, Wolfgang Paul Schmid (a student of Krahe), addressed the first problem by assuming that Krahe's Old European was Proto-Indo-European itself, which he localized in central Europe. This is widely rejected today as few linguists nowadays consider a central European homeland of PIE likely, and it doesn't solve the second problem.
The second problem was addressed by Theo Vennemann (again, a German linguist) who proposed that Old European was not an Indo-European language at all, but related to Basque. Vennemann produced some twenty scholarly papers on this hypothesis, which are collected in the volume Europa Vasconica - Europa Semitica (de Gruyter, 2003).
According to Vennemann, 'Vasconic' languages were spoken all across Europe west of a line that approximately runs from Riga to northern Dalmatia. (East of that line, Uralic was spoken in the north and Indo-European in the south; in the British Isles, Vennemann assumes, an Afro-Asiatic language was spoken, but at an even earlier time, the British Isles, which take part in the Old European hydronymy, had also been Vasconic-speaking.)
However, Vennemann's Vasconic etymologies are very weak and contrived, and the distribution of the Old European river names shows a gap in the area between the Garonne and Ebro rivers - exactly the only area where we know that Basque or something related to it was ever spoken.
A closer inspection of Vennemann's work reveals a number of methodological problems. For instance, he assumes that Europe north of the Alps was uninhabited by humans before the end of the last Ice Age. This assumption is wrong; evidence for human habitation in central Europe during the Ice Age was found at Ahrensburg north-east of Hamburg. In light of this, a linguistic uniformity of prehistoric Europe north of the Alps seems unlikely. Also, it is not generally true that, as Krahe assumes, cities are older than the surrounding villages. There are many known cases of cities which are younger than the surrounding villages. Vennemann also shows a strong tendency towards re-etymologizing semantically transparent names such as Ebersberg and Bischofsheim (both names refer to towns in Bavaria) as "Vasconic".
An important problem is that he manages to reconcile the evidence of Old European place names with Basque data only by brute force. The Old European names do not at all look like Basque, and he has to take recourse to such unparsimonious assumptions as an Italic intermediate substratum in Germany and the British Isles(!) to explain the typological mismatch between Basque and the Old European names.
Rhiemeier's (and others') theory
Jörg Rhiemeier speculated that the names may be from an early diverging branch of Indo-European which he calls Aquan; similar ideas have been formulated by academic scholars such as Francisco Rodríguez Adrados, Francisco Villar or Peter Anreiter. The /a/-centred vocalism of the unknown language possibly preserves a state that existed in Proto-Indo-European before the rise of ablaut (or, alternatively, ablaut was levelled by sound changes). As for the moment, this is just a personal speculation not sufficiently buttressed by solid facts, and Rhiemeier admits that this may be misguided (see next section), using it only as a background for conlangs.
What the competing theories show is that there is more than one way to interpret these names. Clearly, a method that yields such false positives is unreliable. The problem lies in the simple fact that the original meanings of these names are unknown, and we can only speculate about their meanings. This means that we have only resemblances in form without resemblances in meaning, which makes definite statements about the etymologies of the names impossible. Also, we cannot even say whether all the names originate from the same language (or language family) or not. It may very well be the case that the Old European hydronymy is the linguistic equivalent of ley lines - a meaningless pattern falling out of the sheer mass of data points adduced.